- Email: [email protected]
On cyclic codes over Galois rings
Discrete Applied Mathematics (
)
–
Contents lists available at ScienceDirect
Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
On cyclic codes over Galois rings Jasbir Kaur 1 , Sucheta Dutt, Ranjeet Sehmi * Department of Applied Sciences, PEC University of Technology, Chandigarh, India
article
info
Article history: Received 21 March 2016 Received in revised form 11 December 2017 Accepted 20 January 2018 Available online xxxx Keywords: Galois ring Cyclic codes Gröbner basis Minimal degree polynomial Torsion codes
a b s t r a c t Let R be a Galois ring of characteristic pa , where p is a prime and a is a natural number. In this paper, the generators of cyclic codes of arbitrary length n over R in terms of minimal degree polynomials of certain subsets of codes have been obtained. Moreover, the explicit set of generators so obtained turns out to be a minimal strong Gröbner basis. Some results on torsion codes of a cyclic code over R have also been obtained. Using these results, the size of a cyclic code over R has been expressed in terms of the degrees of its generators. © 2018 Elsevier B.V. All rights reserved.
1. Introduction Cyclic codes over finite rings are being studied extensively these days, and the literature is abundant with results on cyclic codes over finite rings where the characteristic s of the ring under consideration and the length n of the code are coprime. For reference see [4,5,9,14,15]. The methodology used in most of these papers is to focus on irreducible factors of xn − 1 and to obtain in turn, the ideals of the ring R[x]/⟨xn − 1⟩ by Hensel’s lifting. However, this technique cannot be applied to codes of general length n as the ring ceases to be a unique factorization domain in case the length of the code and the characteristic s of the ring are not coprime. A few expositions are available for the study of cyclic codes over finite rings in case (n, s) ̸ = 1. For reference see [6,7,10,11,16,17,19]. Dougherty et al. in [7] have given a structure theorem for codes over Galois rings and employed Chinese remainder theorem and lifting of irreducible polynomials. Sălăgean in [16] has given an existential proof for the existence of a minimal strong Gröbner basis for cyclic codes of arbitrary length over a finite chain ring. Norton et al. in [13,14] have formalized the notion of generating set in standard form for cyclic codes over principal ideal ring and obtained necessary and sufficient conditions for the generating set to be a minimal strong Gröbner basis as defined in [2]. Abualrub et al. in [1] have given a simpler approach by introducing minimal degree polynomials to find the generators of cyclic codes of length 2k over Z4 . Garg et al. [8] have extended the approach of Abualrub et al. [1] to cyclic codes over a class of integer residue rings. In this paper, we take further the approach of Abualrub et al. in [1] and find the generators of cyclic codes of general length over Galois rings in an explicit constructive manner, in contrast to the existential results given by Sălăgean in [16]. Also, the set of generators obtained turns out to be a minimal strong Gröbner basis. Further, using our construction we have proved some results about the torsion codes [7] of a cyclic code over a Galois ring. Using these results, the size of a cyclic code over a Galois ring has been obtained in terms of the degrees of its generators.
*
Corresponding author. E-mail addresses: [email protected] (J. Kaur), [email protected] (S. Dutt), [email protected] (R. Sehmi).
1 Work submitted in partial fulfillment of requirements for the degree of Doctor of Philosophy. https://doi.org/10.1016/j.dam.2018.01.017 0166-218X/© 2018 Elsevier B.V. All rights reserved.
Please cite this article in press as: J. Kaur, et al., On cyclic codes over Galois rings, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.01.017.
2
J. Kaur et al. / Discrete Applied Mathematics (
)
–
2. Preliminaries A cyclic code over a ring R is a linear code which is closed under ⟨ cyclic⟩ shifts. It is well known that the cyclic codes of length n over a ring R are in correspondence with the ideals of R[x]/ xn − 1 and thus cyclic codes over R, written as vectors, can be recognized as polynomials of degree less than n, that is, c = (c0 , c1 , . . . , cn−1 ) is identified with the polynomial c0 + c1 x + · · · + cn−1 xn−1 . ⟨ ⟩ A finite ring with identity is called a Galois ring if its zero divisors including zero form a principal ideal p for some prime ⟨ ⟩ p [18]. For any m ≥ 1, the Galois extension ring of Zpa can be constructed as GR(pa , m) = Zpa [x]/ f (x) , where p is a prime, a is a natural number and f (x) ∈ Zpa [x] is a monic basic irreducible polynomial of degree m. The ring GR(pa , m) is called a Galois ring and has pam elements. For a = 1, we obtain the finite field GF (pm ) with pm elements [12,18]. Let I be an ideal in R[x] and A(x) be an element of I . Let lm(A(x)) and lt(A(x)) denote the leading monomial and the leading term, respectively, of A(x). A set G = {Bi (x), 1 ≤ i ≤ ν} of non zero elements of I is called a Gröbner basis of I if for each A(x) ∈ I there exists an i ∈ {1, 2, . . . , ν} such that lm(A(x)) is divisible by lm(Bi (x)). An arbitrary subset G of R[x] is called a Gröbner basis if it is a Gröbner basis of ideal ⟨G⟩. Further, the set G is a strong Gröbner basis for I = ⟨B1 (x), B2 (x), . . . , Bν (x)⟩ if for each A(x) ∈ I , there exists an i ∈ {1, 2, . . . , ν} such that lt(A(x)) is divisible by lt(Bi (x)) [3,2]. Let R be an artinian chain ring, p be the generator of the maximal ideal of R and a be its nilpotency index. Sălăgean [16, Theorem 4.1] has proved that for an artinian chain ring R there exists a Gröbner basis for every ideal of R[x]. For completeness and ready reference, an adaptation of Theorem 3.2 of [14] and Theorem 4.1 of [16] is given below. Theorem 2.1 ([14, Theorem 3.2], [16, Theorem 4.1]). Let R be an artinian chain ring, p be the generator of the maximal ideal of R and a be its nilpotency index. Let G ⊂ R[x] \ {0}. Then G is a minimal strong Gröbner basis if and only if G = {pj0 g0 (x), pj1 g1 (x), . . . , pjs gs (x)} for some s ≤ a − 1 where 1. 2. 3. 4.
0 ≤ j0 < · · · < js ≤ a − 1, gi (x) is a monic polynomial in R[x] for i = 0, 1, . . . , s, n > deg(g0 (x)) > deg(g1 (x)) > · · · > deg(gs (x)), pji+1 gi (x) ∈ ⟨pji+1 gi+1 (x), . . . , pjs gs (x)⟩ in R[x], for i = 0, . . . , s − 1.
3. Generators of cyclic codes over a Galois ring R as ideals of R [x]/⟨xn − 1⟩ Let R = GR(pa , m) be a Galois ring and Rn = R[x]/ xn − 1 . In this section, we find the generators of cyclic codes over R as ideals of Rn . These generators are found in terms of minimal degree polynomials of certain subsets of the given code. Let C be an ideal in Rn and ge (x) be a minimal degree polynomial in C with minimum power of p in the leading coefficient. Let the leading coefficient of ge (x) be pie ue where ue is a unit and 0 ≤ ie ≤ a − 1. If ie = 0 then ge (x) is a monic polynomial. In case ie ̸ = 0, consider a subset Ce−1 of C consisting of polynomials having power of p in the leading coefficient less than ie . Let ge−1 (x) be a minimal degree polynomial in Ce−1 having minimum power of p in the leading coefficient. Let ie−1 denote the power of p in the leading coefficient of ge−1 (x). If ie−1 = 0 then ge−1 (x) is a monic polynomial. If ie−1 ̸ = 0 then successively define the polynomials gj (x) for e − 2 ≥ j ≥ 0 as follows. Consider a subset Cj of C consisting of polynomials having power of p in the leading coefficient less than ij+1 , where ij+1 is the power of p in the leading coefficient of gj+1 (x). Let gj (x) be a minimal degree polynomial in Cj having minimum power of p in the leading coefficient. Let ij denote the power of p in the leading coefficient of gj (x). Note that i0 is the minimum power of p in the leading coefficients among all polynomials in C . Then 0 ≤ i0 < i1 < · · · < ij < ij+1 < · · · < ie . For i0 = 0, g0 (x) is a monic polynomial. Let tj be the degree of the polynomial gj (x). Clearly tj > tj+1 .
⟨
⟩
Remark 1. It is easy to see that for any polynomial c(x) in C with power of p in the leading coefficient l, there exists a smallest j with 0 ≤ j ≤ e such that deg(c(x)) ≥ tj . Then l ≥ ij and the polynomial r(x) = c(x) − pl−ij gj (x)uxdeg(c(x))−tj is in C for some unit u. Moreover, r(x) = 0 or deg(r(x)) < deg(c(x)). The polynomial r(x) can be expressed as r(x) = c(x) − q(x)gj (x) for some q(x) ∈ Rn . The following theorem gives the generators of a cyclic code over the ring R. Theorem 3.1. Let C be an ideal in Rn and gj (x) be polynomials as defined above. Then C = g0 (x), g1 (x), . . . , ge (x) .
⟨
⟩
Proof. Let c(x) be a polynomial in C . By Remark 1, there exists a smallest j and a polynomial q1 (x) ∈ Rn such that the polynomial r1 (x) = c(x) − q1 (x)gj (x) is in( C . Moreover, ) ( r1 (x) ) = 0 or deg r1 (x) < deg c(x) . If r1 (x) = 0 then c(x) ∈ gj (x) ⊂ g0 (x), g1 (x), . . . , ge (x) . If deg r1 (x) < deg c(x) then by Remark 1, there exists a k and a polynomial q2 (x) ∈ Rn such that the polynomial
(
)
(
)
⟨
⟩
⟨
⟩
r2 (x) = r1 (x) − q2 (x)gk (x) Please cite this article in press as: J. Kaur, et al., On cyclic codes over Galois rings, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.01.017.
J. Kaur et al. / Discrete Applied Mathematics (
)
–
3
( ) ( ) ( ) is ⟨ in C . Moreover, ⟩ ⟨ r2 (x) = 0 or deg ⟩ r2 (x) (< deg ) r1 (x) ( < deg ) c(x) . Clearly k ≥ j. If r2 (x) = 0 then c(x) belongs to gj (x), gk (x) ⊂ g0 (x), g1 (x), . . . , ge (x) . If deg r2 (x) < deg r1 (x) then it is evident that after repeating the argument a finite number of times we shall have the remainder equal to zero as the degrees of the remainders form a decreasing sequence of natural numbers which by te . Therefore back substituting for the it is clear that any polynomial ⟨ is bounded below ⟩ ( remainders ) c(x) in C ⟨belongs to gj (x), . . . ,⟩ge (x) where j is the smallest value such that deg c(x) ≥ tj for 0 ≤ j ≤ e. Consequently we get C = g0 (x), g1 (x), . . . , ge (x) . □ The following corollary is an immediate consequence of Theorem 3.1. Corollary 3.2. A cyclic code C of arbitrary length n over a Galois ring of characteristic pa is generated by at most k elements, with k = min{a, t + 1}, where t = max{deg(g(x))}, g(x) a generator. Corollary 3.3. A cyclic code C of arbitrary length n over an integer residue ring of characteristic pa is generated by at most k elements, with k = min{a, t + 1}, where t = max{deg(g(x))}, g(x) a generator. Proof. For m = 1, the Galois ring GR(pa , m) is an integer residue ring of characteristic pa .
□
As finite fields are special case of Galois rings with a = 1, we have the following corollary. Corollary 3.4. A cyclic code C of arbitrary length n over a finite field is generated by a single element. ie Theorem ⟨ 3.5.⟩ Let ge (x) be the polynomial as defined above. Then ge (x) = p he (x), where he (x) is a monic polynomial in Re [x]/ xn − 1 , Re is a Galois ring of characteristic pa−ie .
Proof. Let ge (x) = pie ue xte + bte −1 xte −1 +· · ·+ b0 . Suppose bj ̸ ≡ 0 (mod pie ) for some j, where 0 ≤ j ≤ te − 1. Now pa−ie ge (x) ∈ C ie ie and is a polynomial of ⟨ ⟩ degree less than te , a contradiction. Hence bj ≡ 0 (mod p ) for every j. Thus ge (x) = p he (x) where he (x) ∈ Re [x]/ xn − 1 , Re is a Galois ring of characteristic pa−ie . Clearly he (x) is a monic polynomial. □ Theorem 3.6. Let gj (x), 0 ≤ j ≤ e − 1, be the polynomials as defined earlier. Then 1. pij+1 −ij gj (x) ∈ gj+1 (x), gj+2 (x), . . . , ge (x) . ⟨ ⟩ 2. gj (x) = pij hj (x) where hj (x) is a monic polynomial in Rj [x]/ xn − 1 , Rj is a Galois Ring of characteristic pa−ij . ij+2 −ij+1 ), where ij+2 is to be replaced by a for j = e − 1. 3. hj+1 (x)|hj (x) (mod p
⟨
⟩
Proof. Let c(x) = pij+1 −ij gj (x) − gj+1 (x)xtj −tj+1 . Then c(x) is in C and deg(c(x)) < tj . Now proceeding as in Theorem 3.1, it is easy to see that c(x) = pij+1 −ij gj (x) − gj+1 (x)xtj −tj+1 ∈ ⟨gk (x), gk+1 (x), . . . , ge (x)⟩ for some k > j. This further implies that pij+1 −ij gj (x) ∈ ⟨gj+1 (x), gj+2 (x), . . . , ge (x)⟩.
(1)
This completes the proof for part 1 of the theorem. Next, we need to show that gj (x) = pij hj (x)
(2)
⟨ ⟩ for 0 ≤ j ≤ e − 1. From Theorem 3.5, ge (x) = p he (x), where he (x) is a monic polynomial in R [x]/ xn − 1 , Re is a Galois ring of characteristic pa−ie . Suppose ge−1 (x), ge−2 (x), . . . , gj (x) satisfy (2). We will show that gj−1 (x) satisfies (2). From (1) we have ie
e
pij −ij−1 gj−1 (x) ∈ ⟨gj (x), gj+1 (x), . . . , ge (x)⟩. Therefore, there exist polynomials Fj (x), . . . , Fe (x) in Rn such that pij −ij−1 gj−1 (x) = gj (x)Fj (x) + · · · + ge (x)Fe (x)
= pij hj (x)Fj (x) + · · · + pie he (x)Fe (x) = pij K (x) where K (x) = hj (x)Fj (x) +· · ·+ pie −ij he (x)Fe (x) ∈ Rn . Suppose there exists a coefficient gl,j−1 of the polynomial gj−1 (x) such that gl,j−1 ̸ ≡ 0 (mod pij−1 ). Multiplying both sides by pa−ij we get, pa−ij−1 gj−1 (x) = 0, a contradiction. Thus gj−1 (x) = pij−1 hj−1 (x), where hj−1 (x) is a monic polynomial. Therefore by principle of mathematical induction (2) holds for all j. This proves part 2 of the theorem. Please cite this article in press as: J. Kaur, et al., On cyclic codes over Galois rings, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.01.017.
4
J. Kaur et al. / Discrete Applied Mathematics (
)
–
Next, for 1 ≤ k ≤ a − 1, consider the maps
ψk : GR(pa , m) −→ GR(pk , m) defined by
ψk (α ) = α (mod pk ). ψk is a ring homomorphism for all k which can be extended to ⟨ ⟩ ⟨ ⟩ φk : GR(pa , m)[x]/ xn − 1 −→ GR(pk , m)[x]/ xn − 1 by defining
φk (c0 + c1 x + · · · + cn−1 xn−1 ) = ψk (c0 ) + ψk (c1 )x + · · · + ψk (cn−1 )xn−1 . From (1) and (2) we have pij+1 hj (x) ∈ pij+1 hj+1 (x), pij+2 hj+2 (x), . . . , pie he (x) ,
⟨
⟩
which implies pij+1 hj (x) = pij+1 hj+1 (x)Fj+1 (x) + pij+2 hj+2 (x)Fj+2 (x) + · · · + pie he (x)Fe (x), where Fk (x) ∈ Rn for j + 1 ≤ k ≤ e. Therefore pij+1 hj (x) − hj+1 (x)Fj+1 (x) = pij+2 hj+2 (x)Fj+2 (x) + · · · + pie he (x)Fe (x)
(
)
= pij+2 F (x), where F (x) = hj+2 (x)Fj+2 (x) + · · · + pie −ij+2 he (x)Fe (x). Now pij+1 hj (x) − hj+1 (x)Fj+1 (x) − pij+2 −ij+1 F (x) = 0.
(
)
ij+2 −ij+1 It follows that either the polynomial hj (x) − hj+1 (x)Fj+1⟨(x) − p ⟩ ⟨ F (x) is⟩ equal to zero or the power of p in each of its coefficients is greater than or equal to a − ij+1 . As pa−ij+1 ⊂ pij+2 −ij+1 , the coefficients of the polynomial hj (x) − hj+1 (x)Fj+1 (x) − pij+2 −ij+1 F (x) vanish mod pij+2 −ij+1 in both the cases. Thus
( ) φij+2 −ij+1 hj (x) − hj+1 (x)Fj+1 (x) − pij+2 −ij+1 F (x) = 0. As φij+2 −ij+1 is a homomorphism, we have
( ) ( ) ( ) φij+2 −ij+1 hj (x) = φij+2 −ij+1 hj+1 (x)Fj+1 (x) + φij+2 −ij+1 pij+2 −ij+1 F (x) or
( ) ( ) φij+2 −ij+1 hj (x) = φij+2 −ij+1 hj+1 (x)Fj+1 (x) which gives hj+1 (x)|hj (x) (mod pij+2 −ij+1 ). This proves part 3 of the theorem and hence completes the proof of the theorem. □ Theorem 3.7. Let C be an ideal in Rn and gj (x) be polynomials as defined earlier. Then the set {g0 (x), g1 (x), . . . , ge (x)} is a minimal strong Gröbner basis of C . Proof. The result follows as an immediate consequence of Theorems 3.5, 3.6 and Theorem 2.1. □ Some examples of minimal strong Gröbner basis are given below. Example 3.1. Let G = {g0 (x), g1 (x), g2 (x)} where gj (x) = 2j hj (x) for 0 ≤ j ≤ 2 with h0 (x) = x3 + x2 + x + 1, h1 (x) = x2 + 1 and h2 (x) = x + 1. Let C be the cyclic code of length 8 over Z8 generated by G. It is easy to see that g0 (x), g1 (x) and g2 (x) satisfy the conditions of Theorem 3.1. Moreover, x + 1|x2 + 1 over Z2 and x2 + 1|x3 + x2 + x + 1 over Z4 . Also, 4(x2 + 1) ∈ ⟨4(x + 1)⟩ and 2(x3 + x2 + x + 1) ∈ ⟨2(x2 + 1), 4(x + 1)⟩. Therefore by Theorem 3.7, G is a minimal strong Gröbner basis. Example 3.2. Let G1 = {g0 (x), g1 (x)} where gj (x) = 2j hj (x) for 0 ≤ j ≤ 1 with h0 (x) = x5 + x4 + x3 + x2 + x + 1 and h1 (x) = x4 + x2 + 1. Let C1 be the cyclic code of length 6 over Z4 generated by G1 . Then G1 is a minimal strong Gröbner basis. Example 3.3. Let G2 = {g0 (x), g1 (x)} where gj (x) = 2j hj (x) for 0 ≤ j ≤ 1 with h0 (x) = x3 − 1, h1 (x) = x + 1. Then G2 is a minimal strong Gröbner basis for the cyclic code C2 of length 6 over Z4 . Example 3.4. Let G3 = {g0 (x), g1 (x)} where gj (x) = 2j hj (x) for 0 ≤ j ≤ 1 with h0 (x) = x3 + x2 + x + 1 and h1 (x) = x2 + 1. Let C3 be the cyclic code of length 4 over Z4 generated by G3 . Then G3 is a minimal strong Gröbner basis. Example 3.5. Let G4 = {g0 (x), g1 (x)} where gj (x) = 2j hj (x) for 0 ≤ j ≤ 1 with h0 (x) = x2 + 1 and h1 (x) = x + 1. Then G4 is a minimal strong Gröbner basis for the cyclic code C4 of length 4 over Z4 . Please cite this article in press as: J. Kaur, et al., On cyclic codes over Galois rings, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.01.017.
J. Kaur et al. / Discrete Applied Mathematics (
)
–
5
4. Torsion codes In this section, we obtain the torsion codes of a cyclic code C over a Galois Ring R in terms of minimal degree polynomials of certain subsets of C . We recall some definitions and results given by Dougherty et al. in [7] and Kiah et al. in [10]. Consider the map
φ1 :
R[x] Fpm [x] −→ n ⟨xn − 1⟩ ⟨x − 1⟩
defined earlier as φ1 (f (x)) = f (x), where f (x) = f (x) mod p. Clearly, φ1 is a surjective ring homomorphism. For 0 ≤ i ≤ a − 1, the ith-torsion code of a cyclic code C of length n over R is defined as Tori (C ) = {φ1 v (x) | pi v (x) ∈ C , v (x) ∈ Rn }.
(
)
It may be noted that Tori (C ) is a principally generated ideal in Fpm [x]/⟨xn − 1⟩ for each i. Let Tori (C ) = fi (x) for some polynomial fi (x) ∈ Fpm [x]/⟨xn − 1⟩. Then the degree of the polynomial fi (x) is called the ith-torsional degree of C . Dougherty et al. in [7] have used Hensel’s lemma to lift the generators of torsion codes to obtain generators of a class of cyclic codes over Galois rings. Further, Kiah et al. in [10] have shown that the ith-torsional degree of C is equal to the lowest degree among the polynomials in C with leading coefficient pi . However, we have constructed the generators of cyclic codes of arbitrary length over a Galois ring in terms of minimal degree polynomials of certain subsets of codes (Theorem 3.1). Now we use this structure to obtain the the map φ1 . More precisely, ⟨ torsion codes in terms ⟩ of ⟨minimal degree polynomials under ⟩ we show that for a code C = g0 (x), g1 (x), . . . , ge (x) = pi0 h0 (x), pi1 h1 (x), . . . , pie he (x) over R, where gj (x) = pij hj (x),
⟨
⟩
0 ≤ j ≤ e are the polynomials as defined earlier, the torsion code Torij (C ) is generated by hj (x) and the ij th-torsional degree ( ) of C is equal to the degree of hj (x) which is equal to deg gj (x) = tj . ij Lemma 4.1. Let C = g0 (x), g1 (x),( . . . , ge (x) =) pi0 h0 (x), pi1 h1 (x), . . . , pie he (x) be an ideal in Rn where ( )gj (x) =(p hj (x) ) are ij polynomials as defined earlier. If p A(x) + pB(x) ∈ C , where A(x) ∈ Torij (C ) and B(x) ∈ Rn , then deg A(x) ≥ deg hj (x) .
⟨
⟩
⟨
⟩
Proof. Suppose there exists a polynomial c(x) = pij A(x) + pB(x) ∈ C , where A(x) ∈ Torij (C ) and B(x) ∈ Rn , such that ( ) ( ) ( ) ( ) ( ) deg A(x) < deg hj (x) . Using the minimality of deg ( ) ( gj)(x) , it is easy to see that deg B(x) ≥ deg hj (x) . By Remark 1, there exists a smallest k ≤ j such that deg B(x) ≥ deg gk (x) and hence the polynomial
(
)
r(x) = pij +1 B(x) + pij A(x) − q(x)gk (x)
(
)
is in C for some(q(x) )∈ Rn , where ( )the power of p in the leading coefficient of q(x) is ij + 1 + s − ik for some s ≥ 0. Moreover, r(x) = 0 or deg r(x) < deg B(x) . Comparing the power of p in the coefficients of q(x)gk (x) and pij A(x) it is easy to see that r(x) ̸ = 0. Hence we can write r(x) as r(x) = pij A(x) + pB1 (x)
(
)
where B1 (x) = 0 or deg B1 (x) < deg B(x) . If B1 (x) = 0 then we get a contradiction to the minimality of deg gj (x) . Therefore
( ) ( ) ( ) ) ( ) ( ) deg B1 (x) < deg B(x) . As before, deg B1 (x) ≥ deg hj (x) . Replacing c(x) by r(x) and repeating the above argument a (
)
(
number of times we get a polynomial r ′ (x) = pij A(x) + pB′ (x)
(
)
in C , where B′((x) ∈ Rn and)deg B′ (x) < deg hj (x) . This gives a contradiction(to the ) minimality ( )of deg gj (x) . Thus, for a polynomial pij A(x) + pB(x) ∈ C with A(x) ∈ Torij (C ) and B(x) ∈ Rn , we get deg A(x) ≥ deg hj (x) .
(
)
(
)
(
)
Theorem 4.2. Let C = g0 (x), g1 (x), . . . , ge (x) = pi0 h0 (x), pi1 h1 (x), . . . , pie he (x) be an ideal in Rn where gj (x) = pij hj (x) are
⟨
⟩ ⟨
⟨
⟩
polynomials as defined earlier. Then Torij (C ) = hj (x) and ij th-torsional degree of C = deg hj (x) = tj .
⟩
(
)
Proof. As pij hj (x) ∈ C , we have φ1 hj (x) ∈ Torij (C ). That is, hj (x) ∈ Torij (C ). Thus hj (x) ⊆ Torij (C ). ⟨ ⟩ ( ) To show that Torij (C ) ⊆ hj (x) , let A(x) be an arbitrary polynomial in Torij (C ). Then the polynomial pij A(x) + pB(x) ∈ C ( ) ( ) ( ) ( ) for some B(x) ∈ Rn . Using Lemma 4.1, deg A(x) ≥ deg hj (x) . As hj (x) is a monic polynomial, deg hj (x) = deg φ1 (hj (x)) = ( ) ( ) ( ) deg hj (x) deg A(x) ≥ deg hj (x) . Using division algorithm, there exist unique polynomials q(x) and r(x) in ⟨ . Consequently, ⟩ Fpm [x]/ xn − 1 such that
(
)
⟨
⟩
A(x) = hj (x)q(x) + r(x) where r(x) = 0 or deg r(x) < deg hj (x) . Moreover, r(x) ∈ Torij (C ) as Torij (C ) is an ideal in Fpm [x]/ xn − 1 . Therefore, ( ) ( ) ( ) ( ) there exists some r ′ (x) ∈ Rn such that pij r(x) + pr ′ (x) ∈ C . If deg r(x) < deg hj (x) = deg hj (x) , we get a contradiction ⟨ ⟩ ⟨ ⟩ to Lemma 4.1. Therefore r(x) = 0 and hence A(x) ∈ hj (x) . This gives Torij (C ) = hj (x) . Hence ij th-torsional degree of ( ) ( ) C = deg hj (x) = deg hj (x) = tj .
(
)
(
)
⟨
⟩
Please cite this article in press as: J. Kaur, et al., On cyclic codes over Galois rings, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.01.017.
6
J. Kaur et al. / Discrete Applied Mathematics (
)
–
Remark 2. Let C = g0 (x), g1 (x), . . . , ge (x) = pi0 h0 (x), pi1 h1 (x), . . . , pie he (x) be a cyclic code of length n over R where gj (x) = pij hj (x) are polynomials as defined earlier. Let ij+1 − ij = kj for 0 ≤ j ≤ e − 1 and a − ie = ke . Clearly, Tor0 (C ) = Tor1 (C ) = · · · = Tori0 −1 (C ) = ⟨0⟩ = ⟨xn − 1⟩, Torij (C ) = Torij +1 (C ) = · · · = Torij +kj −1 (C ) ⊂ Torij+1 (C ) for 0 ≤ j ≤ e − 1 and
⟨
⟩
⟨
⟩
Torie (C ) = Torie +1 (C ) = · · · = Torie +ke −1 (C ) = Tora−1 (C ). Using Theorem 4.2 and Remark 2, we get the following theorem as an immediate consequence of Theorem 6.2 of [7] and Theorem 2.2 of [10]. Theorem 4.3. Let C = g0 (x), g1 (x), . . . , ge (x) = pi0 h0 (x), pi1 h1 (x), . . . , pie he (x) be a cyclic code of length n over R where gj (x) = pij hj (x) are polynomials as defined earlier. Let ij+1 − ij = kj for 0 ≤ j ≤ e − 1 and a − ie = ke . Then
⟨
⟩
⟨
⟩
|C | = (pm )an−(i0 n+k0 t0 +k1 t1 +···+ke te ) ( ) where tj = deg gj (x) . The redundant generators of a code as reported by Kiah et al. in Remark 2.7 of [10] can easily be eliminated using Theorem 3.1. Moreover, the torsional degrees of the code can also be calculated using Theorem 4.2 and Remark 2. We illustrate this with the help of the following example. Example 4.1. Let C = ⟨(x − 1)2 , 2(x − 1)2 ⟩ be a cyclic code of length 4 over ( ) Z4 reported by Kiah et al. in [10]. We can easily2 2 verify that C does not have a polynomial ⟨ ⟩ of degree less than deg2 (x − 1) = 2. So minimal degree polynomial in C is (x − 1) itself. Hence by Theorem 3.1, C = g0 (x) where g0 (x) = (x − 1) . It may be noted here that the representation given by Kiah et al. in [10] has a redundant generator 2(x − 1)2 , which has been eliminated using Theorem 3.1. ⟨ ⟩ 2 Clearly, Tor0 (C ) = (x − 1) . Using Remark 2, Tor0 (C ) = Tor1 (C ). So the 0th and 1st-torsional degrees of C are both equal to 2. 5. Conclusion A cyclic code of arbitrary length n over a Galois ring of characteristic pa is generated by at most min{a, t + 1} elements, where t = max{deg(g(x))}, g(x) a generator. Moreover, the set of generators so obtained is a minimal strong Gröbner basis of the code. Some results on torsion codes of a cyclic code over a Galois ring have also been obtained. Using these results, the size of a cyclic code over a Galois ring has been expressed in terms of the degrees of its generators. Acknowledgments The author (Jasbir Kaur) gratefully acknowledges the World Bank funded TEQIP-II for the financial support rendered to the author in the form of scholarship. A part of this paper was presented in the conference CALDAM-2016, Kerala, India during February 18–20, 2016 and was published by Springer in LNCS-9602, Algorithms and Discrete Applied Mathematics. The authors also thank the reviewers of LNCS and Discrete Applied Mathematics for their insightful comments that have helped to improve the manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
T. Abualrub, R. Oehmke, Cyclic codes of length 2e over Z4 , Discrete Appl. Math. 128 (1) (2003) 3–9. W. Adams, P. Loustaunau, An Introduction To Gröbner Basis, American Mathematical Society, 1994. E. Byrne, P. Fitzpatrick, Gröbner bases over Galois rings with an application to decoding alternant codes, J. Symbolic Comput. 31 (2001) 565–584. A.R. Calderbank, N.J.A. Sloane, Modular and p-adic cyclic codes, Des. Codes Cryptogr. 6 (1) (1995) 21–35. H.Q. Dinh, S.R. López-Permouth, Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inform. Theory 50 (8) (2004) 1728–1744. S.T. Dougherty, S. Ling, Cyclic codes over Z4 of even length, Des. Codes Cryptogr. 39 (2) (2006) 127–153. S.T. Dougherty, Y.H. Park, On modular cyclic codes, Finite Fields Appl. 13 (2007) 31–57. A. Garg, S. Dutt, Cyclic codes of length 2k over Z2m , Int. J. Eng. Res. Dev. 1 (9) (2012) 34–37. P. Kanwar, S.R. López-Permouth, Cyclic codes over the integers modulo pm , Finite Fields Appl. 3 (4) (1997) 334–352. H.M. Kiah, K.H. Leung, S. Ling, Cyclic codes over GR(p2 , m) of length pk , Finite Fields Appl. 14 (3) (2008) 834–846. S.R. López-Permouth, H. Özadam, F. Özbudak, S. Szabo, Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes, Finite Fields Appl. 19 (1) (2012) 16–38. B.R. McDonald, Finite Rings with Identity, Marcel Dekker, New York, 1974. G.H. Norton, A. Sălăgean, Strong Gröbner bases for polynomials over a principal ideal ring, Bull. Aust. Math. Soc. 64 (3) (2001) 505–528. G.H. Norton, A. Sălăgean, Cyclic codes and minimal strong Gröbner bases over a principal ideal ring, Finite Fields Appl. 9 (2) (2003) 237–249. B.S. Rajan, M.U. Siddiqi, Transform domain characterization of cyclic codes over Zm , Appl. Algebra Engrg. Comm. Comput. 5 (5) (1994) 261–275. A. Sălăgean, Repeated-root cyclic and negacyclic codes over a finite chain ring, Discrete Appl. Math. 154 (2) (2006) 413–419. R. Sobhani, M. Esmaeili, Cyclic and negacyclic codes over the Galois ring GR(p2 , m), Discrete Appl. Math. 157 (13) (2009) 2892–2903. Z.X. Wan, Finite Fields and Galois Rings, World Scientific Publishing Company, 2011. S.S. Woo, Ideals of Zpn [x]/⟨xl − 1⟩, Commun. Korean Math. Soc. 26 (3) (2011) 427–443.
Please cite this article in press as: J. Kaur, et al., On cyclic codes over Galois rings, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.01.017.
)
–
Contents lists available at ScienceDirect
Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
On cyclic codes over Galois rings Jasbir Kaur 1 , Sucheta Dutt, Ranjeet Sehmi * Department of Applied Sciences, PEC University of Technology, Chandigarh, India
article
info
Article history: Received 21 March 2016 Received in revised form 11 December 2017 Accepted 20 January 2018 Available online xxxx Keywords: Galois ring Cyclic codes Gröbner basis Minimal degree polynomial Torsion codes
a b s t r a c t Let R be a Galois ring of characteristic pa , where p is a prime and a is a natural number. In this paper, the generators of cyclic codes of arbitrary length n over R in terms of minimal degree polynomials of certain subsets of codes have been obtained. Moreover, the explicit set of generators so obtained turns out to be a minimal strong Gröbner basis. Some results on torsion codes of a cyclic code over R have also been obtained. Using these results, the size of a cyclic code over R has been expressed in terms of the degrees of its generators. © 2018 Elsevier B.V. All rights reserved.
1. Introduction Cyclic codes over finite rings are being studied extensively these days, and the literature is abundant with results on cyclic codes over finite rings where the characteristic s of the ring under consideration and the length n of the code are coprime. For reference see [4,5,9,14,15]. The methodology used in most of these papers is to focus on irreducible factors of xn − 1 and to obtain in turn, the ideals of the ring R[x]/⟨xn − 1⟩ by Hensel’s lifting. However, this technique cannot be applied to codes of general length n as the ring ceases to be a unique factorization domain in case the length of the code and the characteristic s of the ring are not coprime. A few expositions are available for the study of cyclic codes over finite rings in case (n, s) ̸ = 1. For reference see [6,7,10,11,16,17,19]. Dougherty et al. in [7] have given a structure theorem for codes over Galois rings and employed Chinese remainder theorem and lifting of irreducible polynomials. Sălăgean in [16] has given an existential proof for the existence of a minimal strong Gröbner basis for cyclic codes of arbitrary length over a finite chain ring. Norton et al. in [13,14] have formalized the notion of generating set in standard form for cyclic codes over principal ideal ring and obtained necessary and sufficient conditions for the generating set to be a minimal strong Gröbner basis as defined in [2]. Abualrub et al. in [1] have given a simpler approach by introducing minimal degree polynomials to find the generators of cyclic codes of length 2k over Z4 . Garg et al. [8] have extended the approach of Abualrub et al. [1] to cyclic codes over a class of integer residue rings. In this paper, we take further the approach of Abualrub et al. in [1] and find the generators of cyclic codes of general length over Galois rings in an explicit constructive manner, in contrast to the existential results given by Sălăgean in [16]. Also, the set of generators obtained turns out to be a minimal strong Gröbner basis. Further, using our construction we have proved some results about the torsion codes [7] of a cyclic code over a Galois ring. Using these results, the size of a cyclic code over a Galois ring has been obtained in terms of the degrees of its generators.
*
Corresponding author. E-mail addresses: [email protected] (J. Kaur), [email protected] (S. Dutt), [email protected] (R. Sehmi).
1 Work submitted in partial fulfillment of requirements for the degree of Doctor of Philosophy. https://doi.org/10.1016/j.dam.2018.01.017 0166-218X/© 2018 Elsevier B.V. All rights reserved.
Please cite this article in press as: J. Kaur, et al., On cyclic codes over Galois rings, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.01.017.
2
J. Kaur et al. / Discrete Applied Mathematics (
)
–
2. Preliminaries A cyclic code over a ring R is a linear code which is closed under ⟨ cyclic⟩ shifts. It is well known that the cyclic codes of length n over a ring R are in correspondence with the ideals of R[x]/ xn − 1 and thus cyclic codes over R, written as vectors, can be recognized as polynomials of degree less than n, that is, c = (c0 , c1 , . . . , cn−1 ) is identified with the polynomial c0 + c1 x + · · · + cn−1 xn−1 . ⟨ ⟩ A finite ring with identity is called a Galois ring if its zero divisors including zero form a principal ideal p for some prime ⟨ ⟩ p [18]. For any m ≥ 1, the Galois extension ring of Zpa can be constructed as GR(pa , m) = Zpa [x]/ f (x) , where p is a prime, a is a natural number and f (x) ∈ Zpa [x] is a monic basic irreducible polynomial of degree m. The ring GR(pa , m) is called a Galois ring and has pam elements. For a = 1, we obtain the finite field GF (pm ) with pm elements [12,18]. Let I be an ideal in R[x] and A(x) be an element of I . Let lm(A(x)) and lt(A(x)) denote the leading monomial and the leading term, respectively, of A(x). A set G = {Bi (x), 1 ≤ i ≤ ν} of non zero elements of I is called a Gröbner basis of I if for each A(x) ∈ I there exists an i ∈ {1, 2, . . . , ν} such that lm(A(x)) is divisible by lm(Bi (x)). An arbitrary subset G of R[x] is called a Gröbner basis if it is a Gröbner basis of ideal ⟨G⟩. Further, the set G is a strong Gröbner basis for I = ⟨B1 (x), B2 (x), . . . , Bν (x)⟩ if for each A(x) ∈ I , there exists an i ∈ {1, 2, . . . , ν} such that lt(A(x)) is divisible by lt(Bi (x)) [3,2]. Let R be an artinian chain ring, p be the generator of the maximal ideal of R and a be its nilpotency index. Sălăgean [16, Theorem 4.1] has proved that for an artinian chain ring R there exists a Gröbner basis for every ideal of R[x]. For completeness and ready reference, an adaptation of Theorem 3.2 of [14] and Theorem 4.1 of [16] is given below. Theorem 2.1 ([14, Theorem 3.2], [16, Theorem 4.1]). Let R be an artinian chain ring, p be the generator of the maximal ideal of R and a be its nilpotency index. Let G ⊂ R[x] \ {0}. Then G is a minimal strong Gröbner basis if and only if G = {pj0 g0 (x), pj1 g1 (x), . . . , pjs gs (x)} for some s ≤ a − 1 where 1. 2. 3. 4.
0 ≤ j0 < · · · < js ≤ a − 1, gi (x) is a monic polynomial in R[x] for i = 0, 1, . . . , s, n > deg(g0 (x)) > deg(g1 (x)) > · · · > deg(gs (x)), pji+1 gi (x) ∈ ⟨pji+1 gi+1 (x), . . . , pjs gs (x)⟩ in R[x], for i = 0, . . . , s − 1.
3. Generators of cyclic codes over a Galois ring R as ideals of R [x]/⟨xn − 1⟩ Let R = GR(pa , m) be a Galois ring and Rn = R[x]/ xn − 1 . In this section, we find the generators of cyclic codes over R as ideals of Rn . These generators are found in terms of minimal degree polynomials of certain subsets of the given code. Let C be an ideal in Rn and ge (x) be a minimal degree polynomial in C with minimum power of p in the leading coefficient. Let the leading coefficient of ge (x) be pie ue where ue is a unit and 0 ≤ ie ≤ a − 1. If ie = 0 then ge (x) is a monic polynomial. In case ie ̸ = 0, consider a subset Ce−1 of C consisting of polynomials having power of p in the leading coefficient less than ie . Let ge−1 (x) be a minimal degree polynomial in Ce−1 having minimum power of p in the leading coefficient. Let ie−1 denote the power of p in the leading coefficient of ge−1 (x). If ie−1 = 0 then ge−1 (x) is a monic polynomial. If ie−1 ̸ = 0 then successively define the polynomials gj (x) for e − 2 ≥ j ≥ 0 as follows. Consider a subset Cj of C consisting of polynomials having power of p in the leading coefficient less than ij+1 , where ij+1 is the power of p in the leading coefficient of gj+1 (x). Let gj (x) be a minimal degree polynomial in Cj having minimum power of p in the leading coefficient. Let ij denote the power of p in the leading coefficient of gj (x). Note that i0 is the minimum power of p in the leading coefficients among all polynomials in C . Then 0 ≤ i0 < i1 < · · · < ij < ij+1 < · · · < ie . For i0 = 0, g0 (x) is a monic polynomial. Let tj be the degree of the polynomial gj (x). Clearly tj > tj+1 .
⟨
⟩
Remark 1. It is easy to see that for any polynomial c(x) in C with power of p in the leading coefficient l, there exists a smallest j with 0 ≤ j ≤ e such that deg(c(x)) ≥ tj . Then l ≥ ij and the polynomial r(x) = c(x) − pl−ij gj (x)uxdeg(c(x))−tj is in C for some unit u. Moreover, r(x) = 0 or deg(r(x)) < deg(c(x)). The polynomial r(x) can be expressed as r(x) = c(x) − q(x)gj (x) for some q(x) ∈ Rn . The following theorem gives the generators of a cyclic code over the ring R. Theorem 3.1. Let C be an ideal in Rn and gj (x) be polynomials as defined above. Then C = g0 (x), g1 (x), . . . , ge (x) .
⟨
⟩
Proof. Let c(x) be a polynomial in C . By Remark 1, there exists a smallest j and a polynomial q1 (x) ∈ Rn such that the polynomial r1 (x) = c(x) − q1 (x)gj (x) is in( C . Moreover, ) ( r1 (x) ) = 0 or deg r1 (x) < deg c(x) . If r1 (x) = 0 then c(x) ∈ gj (x) ⊂ g0 (x), g1 (x), . . . , ge (x) . If deg r1 (x) < deg c(x) then by Remark 1, there exists a k and a polynomial q2 (x) ∈ Rn such that the polynomial
(
)
(
)
⟨
⟩
⟨
⟩
r2 (x) = r1 (x) − q2 (x)gk (x) Please cite this article in press as: J. Kaur, et al., On cyclic codes over Galois rings, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.01.017.
J. Kaur et al. / Discrete Applied Mathematics (
)
–
3
( ) ( ) ( ) is ⟨ in C . Moreover, ⟩ ⟨ r2 (x) = 0 or deg ⟩ r2 (x) (< deg ) r1 (x) ( < deg ) c(x) . Clearly k ≥ j. If r2 (x) = 0 then c(x) belongs to gj (x), gk (x) ⊂ g0 (x), g1 (x), . . . , ge (x) . If deg r2 (x) < deg r1 (x) then it is evident that after repeating the argument a finite number of times we shall have the remainder equal to zero as the degrees of the remainders form a decreasing sequence of natural numbers which by te . Therefore back substituting for the it is clear that any polynomial ⟨ is bounded below ⟩ ( remainders ) c(x) in C ⟨belongs to gj (x), . . . ,⟩ge (x) where j is the smallest value such that deg c(x) ≥ tj for 0 ≤ j ≤ e. Consequently we get C = g0 (x), g1 (x), . . . , ge (x) . □ The following corollary is an immediate consequence of Theorem 3.1. Corollary 3.2. A cyclic code C of arbitrary length n over a Galois ring of characteristic pa is generated by at most k elements, with k = min{a, t + 1}, where t = max{deg(g(x))}, g(x) a generator. Corollary 3.3. A cyclic code C of arbitrary length n over an integer residue ring of characteristic pa is generated by at most k elements, with k = min{a, t + 1}, where t = max{deg(g(x))}, g(x) a generator. Proof. For m = 1, the Galois ring GR(pa , m) is an integer residue ring of characteristic pa .
□
As finite fields are special case of Galois rings with a = 1, we have the following corollary. Corollary 3.4. A cyclic code C of arbitrary length n over a finite field is generated by a single element. ie Theorem ⟨ 3.5.⟩ Let ge (x) be the polynomial as defined above. Then ge (x) = p he (x), where he (x) is a monic polynomial in Re [x]/ xn − 1 , Re is a Galois ring of characteristic pa−ie .
Proof. Let ge (x) = pie ue xte + bte −1 xte −1 +· · ·+ b0 . Suppose bj ̸ ≡ 0 (mod pie ) for some j, where 0 ≤ j ≤ te − 1. Now pa−ie ge (x) ∈ C ie ie and is a polynomial of ⟨ ⟩ degree less than te , a contradiction. Hence bj ≡ 0 (mod p ) for every j. Thus ge (x) = p he (x) where he (x) ∈ Re [x]/ xn − 1 , Re is a Galois ring of characteristic pa−ie . Clearly he (x) is a monic polynomial. □ Theorem 3.6. Let gj (x), 0 ≤ j ≤ e − 1, be the polynomials as defined earlier. Then 1. pij+1 −ij gj (x) ∈ gj+1 (x), gj+2 (x), . . . , ge (x) . ⟨ ⟩ 2. gj (x) = pij hj (x) where hj (x) is a monic polynomial in Rj [x]/ xn − 1 , Rj is a Galois Ring of characteristic pa−ij . ij+2 −ij+1 ), where ij+2 is to be replaced by a for j = e − 1. 3. hj+1 (x)|hj (x) (mod p
⟨
⟩
Proof. Let c(x) = pij+1 −ij gj (x) − gj+1 (x)xtj −tj+1 . Then c(x) is in C and deg(c(x)) < tj . Now proceeding as in Theorem 3.1, it is easy to see that c(x) = pij+1 −ij gj (x) − gj+1 (x)xtj −tj+1 ∈ ⟨gk (x), gk+1 (x), . . . , ge (x)⟩ for some k > j. This further implies that pij+1 −ij gj (x) ∈ ⟨gj+1 (x), gj+2 (x), . . . , ge (x)⟩.
(1)
This completes the proof for part 1 of the theorem. Next, we need to show that gj (x) = pij hj (x)
(2)
⟨ ⟩ for 0 ≤ j ≤ e − 1. From Theorem 3.5, ge (x) = p he (x), where he (x) is a monic polynomial in R [x]/ xn − 1 , Re is a Galois ring of characteristic pa−ie . Suppose ge−1 (x), ge−2 (x), . . . , gj (x) satisfy (2). We will show that gj−1 (x) satisfies (2). From (1) we have ie
e
pij −ij−1 gj−1 (x) ∈ ⟨gj (x), gj+1 (x), . . . , ge (x)⟩. Therefore, there exist polynomials Fj (x), . . . , Fe (x) in Rn such that pij −ij−1 gj−1 (x) = gj (x)Fj (x) + · · · + ge (x)Fe (x)
= pij hj (x)Fj (x) + · · · + pie he (x)Fe (x) = pij K (x) where K (x) = hj (x)Fj (x) +· · ·+ pie −ij he (x)Fe (x) ∈ Rn . Suppose there exists a coefficient gl,j−1 of the polynomial gj−1 (x) such that gl,j−1 ̸ ≡ 0 (mod pij−1 ). Multiplying both sides by pa−ij we get, pa−ij−1 gj−1 (x) = 0, a contradiction. Thus gj−1 (x) = pij−1 hj−1 (x), where hj−1 (x) is a monic polynomial. Therefore by principle of mathematical induction (2) holds for all j. This proves part 2 of the theorem. Please cite this article in press as: J. Kaur, et al., On cyclic codes over Galois rings, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.01.017.
4
J. Kaur et al. / Discrete Applied Mathematics (
)
–
Next, for 1 ≤ k ≤ a − 1, consider the maps
ψk : GR(pa , m) −→ GR(pk , m) defined by
ψk (α ) = α (mod pk ). ψk is a ring homomorphism for all k which can be extended to ⟨ ⟩ ⟨ ⟩ φk : GR(pa , m)[x]/ xn − 1 −→ GR(pk , m)[x]/ xn − 1 by defining
φk (c0 + c1 x + · · · + cn−1 xn−1 ) = ψk (c0 ) + ψk (c1 )x + · · · + ψk (cn−1 )xn−1 . From (1) and (2) we have pij+1 hj (x) ∈ pij+1 hj+1 (x), pij+2 hj+2 (x), . . . , pie he (x) ,
⟨
⟩
which implies pij+1 hj (x) = pij+1 hj+1 (x)Fj+1 (x) + pij+2 hj+2 (x)Fj+2 (x) + · · · + pie he (x)Fe (x), where Fk (x) ∈ Rn for j + 1 ≤ k ≤ e. Therefore pij+1 hj (x) − hj+1 (x)Fj+1 (x) = pij+2 hj+2 (x)Fj+2 (x) + · · · + pie he (x)Fe (x)
(
)
= pij+2 F (x), where F (x) = hj+2 (x)Fj+2 (x) + · · · + pie −ij+2 he (x)Fe (x). Now pij+1 hj (x) − hj+1 (x)Fj+1 (x) − pij+2 −ij+1 F (x) = 0.
(
)
ij+2 −ij+1 It follows that either the polynomial hj (x) − hj+1 (x)Fj+1⟨(x) − p ⟩ ⟨ F (x) is⟩ equal to zero or the power of p in each of its coefficients is greater than or equal to a − ij+1 . As pa−ij+1 ⊂ pij+2 −ij+1 , the coefficients of the polynomial hj (x) − hj+1 (x)Fj+1 (x) − pij+2 −ij+1 F (x) vanish mod pij+2 −ij+1 in both the cases. Thus
( ) φij+2 −ij+1 hj (x) − hj+1 (x)Fj+1 (x) − pij+2 −ij+1 F (x) = 0. As φij+2 −ij+1 is a homomorphism, we have
( ) ( ) ( ) φij+2 −ij+1 hj (x) = φij+2 −ij+1 hj+1 (x)Fj+1 (x) + φij+2 −ij+1 pij+2 −ij+1 F (x) or
( ) ( ) φij+2 −ij+1 hj (x) = φij+2 −ij+1 hj+1 (x)Fj+1 (x) which gives hj+1 (x)|hj (x) (mod pij+2 −ij+1 ). This proves part 3 of the theorem and hence completes the proof of the theorem. □ Theorem 3.7. Let C be an ideal in Rn and gj (x) be polynomials as defined earlier. Then the set {g0 (x), g1 (x), . . . , ge (x)} is a minimal strong Gröbner basis of C . Proof. The result follows as an immediate consequence of Theorems 3.5, 3.6 and Theorem 2.1. □ Some examples of minimal strong Gröbner basis are given below. Example 3.1. Let G = {g0 (x), g1 (x), g2 (x)} where gj (x) = 2j hj (x) for 0 ≤ j ≤ 2 with h0 (x) = x3 + x2 + x + 1, h1 (x) = x2 + 1 and h2 (x) = x + 1. Let C be the cyclic code of length 8 over Z8 generated by G. It is easy to see that g0 (x), g1 (x) and g2 (x) satisfy the conditions of Theorem 3.1. Moreover, x + 1|x2 + 1 over Z2 and x2 + 1|x3 + x2 + x + 1 over Z4 . Also, 4(x2 + 1) ∈ ⟨4(x + 1)⟩ and 2(x3 + x2 + x + 1) ∈ ⟨2(x2 + 1), 4(x + 1)⟩. Therefore by Theorem 3.7, G is a minimal strong Gröbner basis. Example 3.2. Let G1 = {g0 (x), g1 (x)} where gj (x) = 2j hj (x) for 0 ≤ j ≤ 1 with h0 (x) = x5 + x4 + x3 + x2 + x + 1 and h1 (x) = x4 + x2 + 1. Let C1 be the cyclic code of length 6 over Z4 generated by G1 . Then G1 is a minimal strong Gröbner basis. Example 3.3. Let G2 = {g0 (x), g1 (x)} where gj (x) = 2j hj (x) for 0 ≤ j ≤ 1 with h0 (x) = x3 − 1, h1 (x) = x + 1. Then G2 is a minimal strong Gröbner basis for the cyclic code C2 of length 6 over Z4 . Example 3.4. Let G3 = {g0 (x), g1 (x)} where gj (x) = 2j hj (x) for 0 ≤ j ≤ 1 with h0 (x) = x3 + x2 + x + 1 and h1 (x) = x2 + 1. Let C3 be the cyclic code of length 4 over Z4 generated by G3 . Then G3 is a minimal strong Gröbner basis. Example 3.5. Let G4 = {g0 (x), g1 (x)} where gj (x) = 2j hj (x) for 0 ≤ j ≤ 1 with h0 (x) = x2 + 1 and h1 (x) = x + 1. Then G4 is a minimal strong Gröbner basis for the cyclic code C4 of length 4 over Z4 . Please cite this article in press as: J. Kaur, et al., On cyclic codes over Galois rings, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.01.017.
J. Kaur et al. / Discrete Applied Mathematics (
)
–
5
4. Torsion codes In this section, we obtain the torsion codes of a cyclic code C over a Galois Ring R in terms of minimal degree polynomials of certain subsets of C . We recall some definitions and results given by Dougherty et al. in [7] and Kiah et al. in [10]. Consider the map
φ1 :
R[x] Fpm [x] −→ n ⟨xn − 1⟩ ⟨x − 1⟩
defined earlier as φ1 (f (x)) = f (x), where f (x) = f (x) mod p. Clearly, φ1 is a surjective ring homomorphism. For 0 ≤ i ≤ a − 1, the ith-torsion code of a cyclic code C of length n over R is defined as Tori (C ) = {φ1 v (x) | pi v (x) ∈ C , v (x) ∈ Rn }.
(
)
It may be noted that Tori (C ) is a principally generated ideal in Fpm [x]/⟨xn − 1⟩ for each i. Let Tori (C ) = fi (x) for some polynomial fi (x) ∈ Fpm [x]/⟨xn − 1⟩. Then the degree of the polynomial fi (x) is called the ith-torsional degree of C . Dougherty et al. in [7] have used Hensel’s lemma to lift the generators of torsion codes to obtain generators of a class of cyclic codes over Galois rings. Further, Kiah et al. in [10] have shown that the ith-torsional degree of C is equal to the lowest degree among the polynomials in C with leading coefficient pi . However, we have constructed the generators of cyclic codes of arbitrary length over a Galois ring in terms of minimal degree polynomials of certain subsets of codes (Theorem 3.1). Now we use this structure to obtain the the map φ1 . More precisely, ⟨ torsion codes in terms ⟩ of ⟨minimal degree polynomials under ⟩ we show that for a code C = g0 (x), g1 (x), . . . , ge (x) = pi0 h0 (x), pi1 h1 (x), . . . , pie he (x) over R, where gj (x) = pij hj (x),
⟨
⟩
0 ≤ j ≤ e are the polynomials as defined earlier, the torsion code Torij (C ) is generated by hj (x) and the ij th-torsional degree ( ) of C is equal to the degree of hj (x) which is equal to deg gj (x) = tj . ij Lemma 4.1. Let C = g0 (x), g1 (x),( . . . , ge (x) =) pi0 h0 (x), pi1 h1 (x), . . . , pie he (x) be an ideal in Rn where ( )gj (x) =(p hj (x) ) are ij polynomials as defined earlier. If p A(x) + pB(x) ∈ C , where A(x) ∈ Torij (C ) and B(x) ∈ Rn , then deg A(x) ≥ deg hj (x) .
⟨
⟩
⟨
⟩
Proof. Suppose there exists a polynomial c(x) = pij A(x) + pB(x) ∈ C , where A(x) ∈ Torij (C ) and B(x) ∈ Rn , such that ( ) ( ) ( ) ( ) ( ) deg A(x) < deg hj (x) . Using the minimality of deg ( ) ( gj)(x) , it is easy to see that deg B(x) ≥ deg hj (x) . By Remark 1, there exists a smallest k ≤ j such that deg B(x) ≥ deg gk (x) and hence the polynomial
(
)
r(x) = pij +1 B(x) + pij A(x) − q(x)gk (x)
(
)
is in C for some(q(x) )∈ Rn , where ( )the power of p in the leading coefficient of q(x) is ij + 1 + s − ik for some s ≥ 0. Moreover, r(x) = 0 or deg r(x) < deg B(x) . Comparing the power of p in the coefficients of q(x)gk (x) and pij A(x) it is easy to see that r(x) ̸ = 0. Hence we can write r(x) as r(x) = pij A(x) + pB1 (x)
(
)
where B1 (x) = 0 or deg B1 (x) < deg B(x) . If B1 (x) = 0 then we get a contradiction to the minimality of deg gj (x) . Therefore
( ) ( ) ( ) ) ( ) ( ) deg B1 (x) < deg B(x) . As before, deg B1 (x) ≥ deg hj (x) . Replacing c(x) by r(x) and repeating the above argument a (
)
(
number of times we get a polynomial r ′ (x) = pij A(x) + pB′ (x)
(
)
in C , where B′((x) ∈ Rn and)deg B′ (x) < deg hj (x) . This gives a contradiction(to the ) minimality ( )of deg gj (x) . Thus, for a polynomial pij A(x) + pB(x) ∈ C with A(x) ∈ Torij (C ) and B(x) ∈ Rn , we get deg A(x) ≥ deg hj (x) .
(
)
(
)
(
)
Theorem 4.2. Let C = g0 (x), g1 (x), . . . , ge (x) = pi0 h0 (x), pi1 h1 (x), . . . , pie he (x) be an ideal in Rn where gj (x) = pij hj (x) are
⟨
⟩ ⟨
⟨
⟩
polynomials as defined earlier. Then Torij (C ) = hj (x) and ij th-torsional degree of C = deg hj (x) = tj .
⟩
(
)
Proof. As pij hj (x) ∈ C , we have φ1 hj (x) ∈ Torij (C ). That is, hj (x) ∈ Torij (C ). Thus hj (x) ⊆ Torij (C ). ⟨ ⟩ ( ) To show that Torij (C ) ⊆ hj (x) , let A(x) be an arbitrary polynomial in Torij (C ). Then the polynomial pij A(x) + pB(x) ∈ C ( ) ( ) ( ) ( ) for some B(x) ∈ Rn . Using Lemma 4.1, deg A(x) ≥ deg hj (x) . As hj (x) is a monic polynomial, deg hj (x) = deg φ1 (hj (x)) = ( ) ( ) ( ) deg hj (x) deg A(x) ≥ deg hj (x) . Using division algorithm, there exist unique polynomials q(x) and r(x) in ⟨ . Consequently, ⟩ Fpm [x]/ xn − 1 such that
(
)
⟨
⟩
A(x) = hj (x)q(x) + r(x) where r(x) = 0 or deg r(x) < deg hj (x) . Moreover, r(x) ∈ Torij (C ) as Torij (C ) is an ideal in Fpm [x]/ xn − 1 . Therefore, ( ) ( ) ( ) ( ) there exists some r ′ (x) ∈ Rn such that pij r(x) + pr ′ (x) ∈ C . If deg r(x) < deg hj (x) = deg hj (x) , we get a contradiction ⟨ ⟩ ⟨ ⟩ to Lemma 4.1. Therefore r(x) = 0 and hence A(x) ∈ hj (x) . This gives Torij (C ) = hj (x) . Hence ij th-torsional degree of ( ) ( ) C = deg hj (x) = deg hj (x) = tj .
(
)
(
)
⟨
⟩
Please cite this article in press as: J. Kaur, et al., On cyclic codes over Galois rings, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.01.017.
6
J. Kaur et al. / Discrete Applied Mathematics (
)
–
Remark 2. Let C = g0 (x), g1 (x), . . . , ge (x) = pi0 h0 (x), pi1 h1 (x), . . . , pie he (x) be a cyclic code of length n over R where gj (x) = pij hj (x) are polynomials as defined earlier. Let ij+1 − ij = kj for 0 ≤ j ≤ e − 1 and a − ie = ke . Clearly, Tor0 (C ) = Tor1 (C ) = · · · = Tori0 −1 (C ) = ⟨0⟩ = ⟨xn − 1⟩, Torij (C ) = Torij +1 (C ) = · · · = Torij +kj −1 (C ) ⊂ Torij+1 (C ) for 0 ≤ j ≤ e − 1 and
⟨
⟩
⟨
⟩
Torie (C ) = Torie +1 (C ) = · · · = Torie +ke −1 (C ) = Tora−1 (C ). Using Theorem 4.2 and Remark 2, we get the following theorem as an immediate consequence of Theorem 6.2 of [7] and Theorem 2.2 of [10]. Theorem 4.3. Let C = g0 (x), g1 (x), . . . , ge (x) = pi0 h0 (x), pi1 h1 (x), . . . , pie he (x) be a cyclic code of length n over R where gj (x) = pij hj (x) are polynomials as defined earlier. Let ij+1 − ij = kj for 0 ≤ j ≤ e − 1 and a − ie = ke . Then
⟨
⟩
⟨
⟩
|C | = (pm )an−(i0 n+k0 t0 +k1 t1 +···+ke te ) ( ) where tj = deg gj (x) . The redundant generators of a code as reported by Kiah et al. in Remark 2.7 of [10] can easily be eliminated using Theorem 3.1. Moreover, the torsional degrees of the code can also be calculated using Theorem 4.2 and Remark 2. We illustrate this with the help of the following example. Example 4.1. Let C = ⟨(x − 1)2 , 2(x − 1)2 ⟩ be a cyclic code of length 4 over ( ) Z4 reported by Kiah et al. in [10]. We can easily2 2 verify that C does not have a polynomial ⟨ ⟩ of degree less than deg2 (x − 1) = 2. So minimal degree polynomial in C is (x − 1) itself. Hence by Theorem 3.1, C = g0 (x) where g0 (x) = (x − 1) . It may be noted here that the representation given by Kiah et al. in [10] has a redundant generator 2(x − 1)2 , which has been eliminated using Theorem 3.1. ⟨ ⟩ 2 Clearly, Tor0 (C ) = (x − 1) . Using Remark 2, Tor0 (C ) = Tor1 (C ). So the 0th and 1st-torsional degrees of C are both equal to 2. 5. Conclusion A cyclic code of arbitrary length n over a Galois ring of characteristic pa is generated by at most min{a, t + 1} elements, where t = max{deg(g(x))}, g(x) a generator. Moreover, the set of generators so obtained is a minimal strong Gröbner basis of the code. Some results on torsion codes of a cyclic code over a Galois ring have also been obtained. Using these results, the size of a cyclic code over a Galois ring has been expressed in terms of the degrees of its generators. Acknowledgments The author (Jasbir Kaur) gratefully acknowledges the World Bank funded TEQIP-II for the financial support rendered to the author in the form of scholarship. A part of this paper was presented in the conference CALDAM-2016, Kerala, India during February 18–20, 2016 and was published by Springer in LNCS-9602, Algorithms and Discrete Applied Mathematics. The authors also thank the reviewers of LNCS and Discrete Applied Mathematics for their insightful comments that have helped to improve the manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
T. Abualrub, R. Oehmke, Cyclic codes of length 2e over Z4 , Discrete Appl. Math. 128 (1) (2003) 3–9. W. Adams, P. Loustaunau, An Introduction To Gröbner Basis, American Mathematical Society, 1994. E. Byrne, P. Fitzpatrick, Gröbner bases over Galois rings with an application to decoding alternant codes, J. Symbolic Comput. 31 (2001) 565–584. A.R. Calderbank, N.J.A. Sloane, Modular and p-adic cyclic codes, Des. Codes Cryptogr. 6 (1) (1995) 21–35. H.Q. Dinh, S.R. López-Permouth, Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inform. Theory 50 (8) (2004) 1728–1744. S.T. Dougherty, S. Ling, Cyclic codes over Z4 of even length, Des. Codes Cryptogr. 39 (2) (2006) 127–153. S.T. Dougherty, Y.H. Park, On modular cyclic codes, Finite Fields Appl. 13 (2007) 31–57. A. Garg, S. Dutt, Cyclic codes of length 2k over Z2m , Int. J. Eng. Res. Dev. 1 (9) (2012) 34–37. P. Kanwar, S.R. López-Permouth, Cyclic codes over the integers modulo pm , Finite Fields Appl. 3 (4) (1997) 334–352. H.M. Kiah, K.H. Leung, S. Ling, Cyclic codes over GR(p2 , m) of length pk , Finite Fields Appl. 14 (3) (2008) 834–846. S.R. López-Permouth, H. Özadam, F. Özbudak, S. Szabo, Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes, Finite Fields Appl. 19 (1) (2012) 16–38. B.R. McDonald, Finite Rings with Identity, Marcel Dekker, New York, 1974. G.H. Norton, A. Sălăgean, Strong Gröbner bases for polynomials over a principal ideal ring, Bull. Aust. Math. Soc. 64 (3) (2001) 505–528. G.H. Norton, A. Sălăgean, Cyclic codes and minimal strong Gröbner bases over a principal ideal ring, Finite Fields Appl. 9 (2) (2003) 237–249. B.S. Rajan, M.U. Siddiqi, Transform domain characterization of cyclic codes over Zm , Appl. Algebra Engrg. Comm. Comput. 5 (5) (1994) 261–275. A. Sălăgean, Repeated-root cyclic and negacyclic codes over a finite chain ring, Discrete Appl. Math. 154 (2) (2006) 413–419. R. Sobhani, M. Esmaeili, Cyclic and negacyclic codes over the Galois ring GR(p2 , m), Discrete Appl. Math. 157 (13) (2009) 2892–2903. Z.X. Wan, Finite Fields and Galois Rings, World Scientific Publishing Company, 2011. S.S. Woo, Ideals of Zpn [x]/⟨xl − 1⟩, Commun. Korean Math. Soc. 26 (3) (2011) 427–443.
Please cite this article in press as: J. Kaur, et al., On cyclic codes over Galois rings, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.01.017.