Lee weights of cyclic self-dual codes over Galois rings of characteristic p2

Finite Fields and Their Applications 45 (2017) 107–130

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Finite Fields and Their Applications www.elsevier.com/locate/ffa

Lee weights of cyclic self-dual codes over Galois rings of characteristic p2 Boran Kim, Yoonjin Lee ∗,1 Department of Mathematics, Ewha Womans University, Seoul 03760, Republic of Korea

a r t i c l e

i n f o

Article history: Received 26 May 2016 Received in revised form 11 October 2016 Accepted 22 November 2016 Available online xxxx Communicated by Chaoping Xing MSC: 94B15

a b s t r a c t We completely determine the minimum Lee weights of cyclic self-dual codes over a Galois ring GR(p2 , m) of length pk , where m and k are positive integers and p is a prime number. We obtain all cyclic self-dual codes over GR(22 , 1) ∼ = Z4 of lengths 16 and 32 with their Lee weight enumerators. We also find cyclic self-dual codes over GR(32 , 1) ∼ = Z9 (respectively, GR(32 , 2)) of lengths up to 27 (respectively, 9). Most of the cyclic self-dual codes we found are extremal with respect to the Lee weights. © 2016 Elsevier Inc. All rights reserved.

Keywords: Cyclic code Self-dual code Galois ring Minimum Lee weight Extremal code

1. Introduction There have been active developments on cyclic self-dual codes over finite rings in terms of their structures and minimum weights (for instance, refer to [3], [4], [5], [7], * Corresponding author. E-mail addresses: [email protected] (B. Kim), [email protected] (Y. Lee). The author is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827) and also by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (2014-002731). 1

http://dx.doi.org/10.1016/j.ffa.2016.11.015 1071-5797/© 2016 Elsevier Inc. All rights reserved.

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[13]). In [14], the authors find cyclic self-dual codes over Z4 of odd lengths and give a list of cyclic self-dual codes of lengths up to 39. In [1], the authors completely determine the generators of cyclic codes over Z4 of length 2k , where k is a positive integer. A ring GR(pe , m) = Zpe [x]/g(x) is called a Galois ring of characteristic pe with (pe )m elements, where p is a prime number and g(x) is a monic basic irreducible polynomial of degree m in Zpe [x] for a positive integer e. A class of Galois rings is a broad class of rings which contains finite rings Zpe ∼ = GR(pe , 1) and finite fields Fpm ∼ = GR(p, m). In [10,15], the authors obtain a mass formula for cyclic self-dual codes over GR(p2 , m) of length pk . In [8], the authors find the minimum Hamming weights of all cyclic codes over Z4 of length 2k and some partial results on the minimum Lee weights of cyclic codes over Z4 of length 2k . In [6], the authors classify generators of cyclic codes over GR(22 , m) of length 2k , and they find necessary conditions for cyclic codes to be self-dual. These results over GR(22 , m) of length 2k are extended to codes over GR(p2 , m) of length pk in [16]. Moreover, in [12], the authors obtain the Hamming weights of cyclic codes over GR(p2 , m) of length pk . Our goal is to explicitly determine the minimum Lee weights of cyclic self-dual codes over a Galois ring GR(p2 , m) of length pk , and we obtain the main result in Theorem 1.1, where m and k are positive integers and p is a prime number. We obtain all cyclic self-dual codes over GR(22 , 1) ∼ = Z4 of lengths 16 and 32 with their Lee weight enumerators. We also find cyclic self-dual codes over GR(32 , 1) ∼ = Z9 (respectively, GR(32 , 2)) of lengths up to 27 (respectively, 9). Most of the cyclic self-dual codes we found are extremal with respect to the Lee weights (Theorems 4.1, 4.2 and 4.3). We use the following notation throughout this paper. Notation p a prime number m, k positive integers a finite field with pm elements Fpm 2 R = GR(p , m) a Galois ring of characteristic p2 with (p2 )m elements R a quotient ring GR(p2 , m)/p, where p is a principal ideal of GR(p2 , m) (in fact, R is isomorphic to the Teichmuller set of coset representatives of GR(p2 , m)) ξ a nonzero element in GR(p2 , m) of order pm − 1 a cyclic code over GR(p2 , m) of length n = pk associated with C an ideal I = (x − 1)r + pf (x), p(x − 1)t  of R[x]/xn − 1, where t−1 r + t ≤ n and f (x) = i=0 ai (x − 1)i with ai ∈ R a a binomial coefficient indexed by a and b b a dual code of C C⊥ t the torsion degree of a cyclic code C r the residue degree of a cyclic code C g(x) an ideal of GR(p2 , m)[x]/xn − 1 generated by g(x) ∗ a reciprocal polynomial of g(x) g (x)

B. Kim, Y. Lee / Finite Fields and Their Applications 45 (2017) 107–130

wtH (v) wtL (v)

109

Hamming weight of a codeword v Lee weight of a codeword v

Any cyclic code over GR(p2 , m) of length n = pk is uniquely associated with an ideal I = (x − 1)r + pf (x), p(x − 1)t  of R[x]/xn − 1, where r (resp. t) is called the residue (resp. torsion) degree, f (x) = t−1 i i=0 ai (x − 1) with ai ∈ R and r + t ≤ n; we call f (x) the associated polynomial of C (this is justified in Lemma 2.3). The following Theorem 1.1 is our main result, and its proof is given in the section 3. Theorem 1.1. Let C be a cyclic self-dual code over GR(p2 , m) of length n = pk associated with an ideal I = (x − 1)r + pf (x), p(x − 1)n−r  of GR(p2 , m)[x]/xn − 1, where t−1 f (x) = i=0 ai (x − 1)i with ai ∈ R and (p − 1)pk−1 ≤ r ≤ n. Case 1. If r = n, then C has the minimum Lee weight p. Case 2. If r = n and p = 2, then C has the minimum Lee weight 4. Case 3. If r = n and p = 2, then we have the following two cases: (i) If r = (p − 1)pk−1 , then C has the minimum Lee weight 2p. (ii) Suppose that r = (p − 1)pk−1 (then I is a principal ideal (x − 1)r + pf (x) with a0 = 0, 1). Then C has the minimum Lee weight 2p − 1 C has the minimum Lee weight 2p

if f (x) = a0 , and if f (x) = a0 .

2. Preliminaries A monic irreducible polynomial g(x) in Zp2 [x] is called a monic basic irreducible polynomial if its reduction modulo p is also an irreducible polynomial in Zp [x]. Let GR(p2 , m) = Zp2 [x]/g(x) be a Galois ring of characteristic p2 with (p2 )m elements, where g(x) is a monic basic irreducible polynomial of degree m in Zp2 [x]. A Galois ring GR(p2 , m) is a local ring with the maximal ideal p, and GR(p2 , m) mod p is isomorphic to the Teichmuller set of coset representatives of R. Moreover, GR(p2 , m) is a finite chain ring and its ideals are pi  with 0 ≤ i ≤ 2. A linear code of length n over GR(p2 , m) is a submodule of GR(p2 , m)n . An element of C is called a codeword. In this paper, we assume that every code is a linear code. We define the inner product of vectors v = (v1 , . . . , vn ) and w = (w1 , . . . , wn ) in GR(p2 , m)n n by v · w = i=1 vi wi . The dual code C ⊥ of C is defined by C ⊥ = {v ∈ GR(p2 , m)n | v · w = 0 for all w ∈ C}. If C ⊆ C ⊥ , then C is called self-orthogonal, and if C = C ⊥ , then C is called self-dual.

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The following proposition shows two possible ways in which we can express elements of GR(p2 , m). Proposition 2.1. ([17, Theorem 14.8]) Let e be a positive integer. (i) GR(pe , m) = Zpe [ξ] = {h0 + h1 ξ + · · · + hm−1 ξ m−1 : h0 , h1 , · · · , hm−1 ∈ Zpe }. (ii) Any element c ∈ GR(pe , m) can be written uniquely as c = a0 + a1 p + · · · + ae−1 pe−1 , where a0 , a1 , · · · , ae−1 ∈ R. Moreover, c is a unit in GR(pe , m) if and only if a0 = 0. In [2, p. 463], the authors define the Lee weight of v over Zpe (e ≥ 1) as follows: wtL (v) = min{v, pe − v}, where p is a prime number. However, there is no distance preserving Gray map under L over Zpe , and this weight. Therefore, in [18], the authors define the Lee weight wt e−1 p L . there exists a distance preserving Gray map from Zpe to Zp under this weight wt L defined over Zpe to Galois rings. In the Furthermore, they extend the Lee weight wt following definition, we just state the case of e = 2. Definition 2.2. ([18, p, 147, p. 151]) (i) For v = (v1 , . . . , vn ) ∈ Znp2 , the Lee weight of v over Zp2 is L (v) = wt

n 

L (vi ), wt

i=1

where

⎧ ⎪ ⎪ ⎨vi  wtL (vi ) = p ⎪ ⎪ ⎩p2 − v

i

if vi ≤ p, if p ≤ vi ≤ p2 − p, if p2 − p < vi ≤ p2 − 1.

(ii) For any u ∈ GR(p2 , m), the Lee weight of u over GR(p2 , m) is wtL (u) =

m−1 

L (ui ), wt

i=0

where u = we define

m−1 i=0

ui ξ i (by Proposition 2.1). So, for v = (v1 , . . . , vn ) in GR(p2 , m)n ,

wtL (v) =

n  i=1

wtL (vi ).

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The Hamming weight wtH (v) of a vector v = (v1 , . . . , vn ) is the number of nonzero components vi in the vector. We note that wtL (v) ≥ wtH (v) for any vector v. The minimum Lee (resp. Hamming) weight of a code C is the smallest Lee (resp. Hamming) weight among nonzero weights of all codewords. A cyclic code C over GR(p2 , m) of length n is a code which is invariant with respect to the shift operator that maps an element (c0 , c1 , . . . , cn−1 ) on GR(p2 , m) to an element (cn−1 , c0 , . . . , cn−2 ). We note that there is the natural correspondence between cyclic codes over GR(p2 , m) of length n = pk and ideals of GR(p2 , m)[x]/xn − 1. For the study of cyclic codes over GR(p2 , m) of length n = pk , we investigate ideals of GR(p2 , m)[x]/xn − 1. From now on, we say that a cyclic code C over GR(p2 , m) of length n = pk is associated with an ideal I of GR(p2 , m)[x]/xn − 1. We define Ti (I) = μ({v ∈ GR(p2 , m)[x]/xn − 1 | pi v ∈ I}) (i = 0, 1), where I is an ideal of GR(p2 , m)[x]/xn − 1 and μ : GR(p2 , m)[x]/xn − 1 → R[x]/xn − 1, f (x) → f (x) (mod p) is a surjective ring homomorphism. A cyclic code C over R of length pk which is associated with an ideal T0 (I) (resp. T1 (I)) of R[x]/xn − 1 is called the residue code (resp. torsion code), denoted by Res(C) (resp. T or(C)). By [9, Theorem 2.2], Res(C) and T or(C) are associated with ideals (x − 1)r  and (x − 1)t  of R[x]/xn − 1 for some 0 ≤ r, t ≤ n, respectively. In this case, r (resp. t) is equal to the residue degree (resp. torsion degree) of C. The following lemma shows that a cyclic code over GR(p2 , m) of length n = pk is uniquely associated with an ideal of GR(p2 , m)[x]/xn − 1. Lemma 2.3. ([9, Theorem 3.8]) Let U be the set of ideals of GR(p2 , m)[x]/xn − 1 with n = pk and I an element of U such that T0 (I) = (x − 1)r  and T1 (I) = (x − 1)t , where r (resp. t) is a residue degree (resp. torsion degree). Let A = {I ∈ U | r + t ≤ pk } and J = {(a, b, a0 , a1 , . . . , ab−1 ) | 0 ≤ a < pk , 0 ≤ b ≤ min{pk−1 , a}, a + b ≤ pk , ai ∈ R}. The map ψ : J → A \ p defined by (a, b, a0 , a1 , . . . , ab−1 ) → (x − 1)a + p

b−1 

ai (x − 1)i , p(x − 1)b 

(1)

i=0

is a bijection map. Moreover, setting I = (x − 1)a + p have r = a and t = b.

b−1 i=0

ai (x − 1)i , p(x − 1)b , we

Throughout this paper, we denote a codeword v = (v0 , . . . , vs ) in C as a polynomial s qi ∈ GR(p2 , m)[x]/xn − 1 (0 ≤ s ≤ n − 1) with 0 ≤ qi ≤ n − 1 and qi = qj i=0 vi x

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(i = j). The Hamming weight (resp. Lee weight) of a codeword is denoted by wtH (v) (resp. wtL (v)), where v = (v0 , . . . , vs ) ∈ GR(p2 , m)s+1 . The following proposition presents the minimum Hamming weight of a cyclic code over R of length n = pk associated with a principal ideal (x − 1)i  of R[x]/xn − 1. Proposition 2.4. ([12, Theorem 7.5]) Let C[i] be a cyclic code over R of length pk which k is associated with an ideal (x − 1)i  of R[x]/xp − 1 for 0 ≤ i ≤ pk − 1. Let dH (C[i]) be the minimum Hamming weight of a code C[i]. If p = 2, then ⎧ ⎪ ⎪ ⎨1 dH (C[i]) = 2 ⎪ ⎪ ⎩2s+1

if i = 0, if 1 ≤ i ≤ 2k−1 , if 2k − 2k−s + 1 ≤ i ≤ 2k − 2k−s + 2k−s−1 , where 1 ≤ s ≤ k − 1.

If p is an odd prime, then ⎧ ⎪ 2 ⎪ ⎪ ⎪ ⎨β + 2 dH (C[i]) = ⎪ (τ + 1)ps ⎪ ⎪ ⎪ ⎩

if 1 ≤ i ≤ pk−1 , if βpk−1 + 1 ≤ i ≤ (β + 1)pk−1 , where 1 ≤ β ≤ p − 2, if pk − pk−s + (τ − 1)pk−s−1 + 1 ≤ i ≤ pk − pk−s + τ pk−s−1 , where 1 ≤ τ ≤ p − 1 and 1 ≤ s ≤ k − 1.

Definition 2.5. Let I be an ideal of R[x]/xn − 1 and h(x) = in R[x]/xn − 1 with n = pk .

n−1 i=0

ai xi a polynomial

(i) The annihilator of I is an ideal A(I) = {h(x) ∈ R[x]/xn − 1 | h(x)g(x) = 0 for all g(x) ∈ I} of R[x]/xn − 1. n−1 (ii) The reciprocal of h(x), denoted by h∗ (x), is a polynomial i=0 ai xn−1−i . For any subset S of R[x]/xn − 1, the set S ∗ is {h∗ (x) | h(x) ∈ S}. The following proposition shows that the dual code C ⊥ of C over GR(p2 , m) of length n = pk is associated with a specific type ideal of R[x]/xn − 1. Proposition 2.6. ([16, Lemma 3.3]) Let C be a cyclic code over GR(p2 , m) of length n = pk associated with an ideal I of R[x]xn − 1. Then C ⊥ is associated with an ideal A(I)∗ of R[x]/xn − 1. 3. Determination of weights of cyclic self-dual codes over GR(p2 , m) of length pk The following proposition shows the classification of explicit generators of cyclic selfdual codes over GR(p2 , m) of length pk . The result follows by combining [6, Theorem 5.3, Proposition 5.5], [16, Corollary 4.4, Lemma 4.5], [15, Theorem 2] and [10].

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Proposition 3.1. Let C be a cyclic code over GR(p2 , m) of length n = pk associated with an ideal I = (x − 1)r + pf (x), p(x − 1)t  of GR(p2 , m)[x]/xn − 1, where f (x) = t−1 i i=0 ai (x − 1) and ai ∈ R (0 ≤ i ≤ t − 1). Let s be the smallest non-negative integer such that as = 0. Then C is self-dual if and only if I belongs to one of the following cases: Case 1. f (x) = 0. Then one of the following cases holds: (i) r = n (I: principal) (ii) r =  n, t = n − r and 2r ≥ 2pk − pk−1 (I: non-principal) n−r

Case 2. f (x) = 0, where (a0 , . . . , an−r−1 ) ∈ R is a nonzero vector which is a solution of the following simultaneous systematic equations: for all 0 ≤  ≤ n − r − 1, n−r−1 i=0

(ηi, − τi, )ai = 0

if 2r ≥ 2n − pk−1 ,

i=0

(ηi, − τi, )ai = −δi

if 2r < 2n − pk−1 ,

n−r−1

i+r+1

where ηi, = (−1) δi =

 1

i

 r−j   , τi, =

j i j=0 (−1) j

 1

if i = 

0

if i = 

(2)

and

if i = r − (p − 1)pk−1

0 if i = r − (p − 1)pk−1 . In addition, one of the following cases holds: (i) s = 0 and (p − 1)pk−1 ≤ r ≤ n − 1, where if r = (p − 1)pk−1 , then we require that p = 2 and a0 = 1 (I: principal). (ii) s = 0, (p − 1)pk−1 < r ≤ n − 1, t = n − r and s ≤ r − (p − 1)pk−1 , where if s = r − (p − 1)pk , then we require that p = 2 and as = 1 (I: non-principal). 3.1. Minimum Hamming weights of cyclic self-dual codes over GR(p2 , m) of length pk In the following proposition, we determine the minimum Hamming weights of cyclic self-dual codes over GR(p2 , m) of length pk . Proposition 3.2. Let C be a cyclic self-dual code over GR(p2 , m) of length n = pk with the residue degree r. Then C has the minimum Hamming weight 1 if r = n and 2 if r = n. Proof. We note that t + r = n since C is self-dual by Proposition 3.1; in particular, for Case 2 (i) of Proposition 3.1, we show that t + r = n in the proof of the case. If r = n, then a cyclic self-dual code C is associated with the ideal p of R[x]/xn − 1. We then have that the minimum Hamming weight of C is equal to 1.

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If not, we have (p − 1)pk−1 ≤ r ≤ n − 1 by Proposition 3.1. Thus, the torsion degree t of C satisfies 1 ≤ t ≤ pk−1 since t + r = n. Then we get that the minimum Hamming weights of C and T or(C) are the same as 2 by [12, Lemma 4.8] and Proposition 2.4. 2 3.2. Minimum Lee weights of cyclic self-dual codes over GR(p2 , m) of length pk We need the following lemma for the proof of our main theorem. Lemma 3.3. (i) Let xs be an element in GR(p2 , m)[x] (s ≥ 1). Then we have xs = k k k (ii) (x − 1)p = xp − 1 in R[x]/xp − 1.

s

s

i=0

i

(x − 1)i .

Proof. (i) By the mathematical induction and Pascal’s rule, the result follows. If s = 1, then τ   it is true. Suppose that it is true when s = τ (that is, xτ = i=0 τi (x − 1)i ). Then we have τ   xτ +1 = x i=0 τi (x − 1)i τ   = (1 + (x − 1)) i=0 τi (x − 1)i τ τ  τ   = i=0 i (x − 1)i + i=0 τi (x − 1)i+1 τ        = 0 (x − 1)0 + τ1 + τ0 (x − 1) + · · · + ττ (x − 1)τ +1     τ +1 τ +1 = τ +1 (x − 1)0 + τ +1 1 (x − 1) + · · · + τ +1 (x − 1) 0τ +1 τ +1 i = i=0 i (x − 1) . So, the result follows. (ii) We have p k  p k

pk

(x − 1)

=

i=0

i

k

xp

−i

k

(−1)i = xp − 1

k

in R[x]/xp − 1. 2 k

The next lemma shows how to determine all the units of R[x]/xp − 1. k

Lemma 3.4. Let h(x) be an element of R[x]/xp − 1, where h(x) can be written by pk −1 pk −1 i i pk − 1 with hij ∈ R (j = 1, 2). i=0 hi (x − 1) = i=0 (hi0 + phi1 )(x − 1) in R[x]/x k Then h(x) is a unit in R[x]/xp − 1 if and only if h00 = 0 in R. k

Proof. We note that R[x]/xp −1 is a local ring with the maximal ideal p, x −1. Every pk −1 pk −1 k element of R[x]/xp − 1 has the form i=0 hi (x − 1)i = i=0 (hi0 + phi1 )(x − 1)i

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with hij ∈ R (j = 1, 2) by Proposition 2.1 (ii) and Lemma 3.3. Thus, h(x) is a unit of k R[x]/xp − 1 if and only if h(x) does not belong to the maximal ideal p, x − 1 of k k R[x]/xp − 1. It thus follows that h(x) is a unit of R[x]/xp − 1 if and only if h00 = 0 in R. 2 In the following lemma, we find an upper bound on the minimum Lee weight of a cyclic self-dual code over GR(p2 , m). Lemma 3.5. Let C be a cyclic self-dual code over GR(p2 , m) of length n = pk with r = n (then we note that C has the minimum Hamming weight 2 by Proposition 3.2). Let v be a codeword with wtH (v) = 2. Then wtL (v) ≥ 2p. Furthermore, there exists v ∈ C v ) = 2p (in fact, v = uv for some unit u ∈ GR(p2 , m)); this implies that the with wtL ( minimum Lee weight of C is less than or equal to 2p. Proof. Let v = v1 xq1 + v2 xq2 be a codeword in C such that wtH (v) = 2 with nonzero v1 , v2 ∈ GR(p2 , m). We first claim that both v1 and v2 are nonunits in GR(p2 , m). Case 1. Suppose that v1 is a unit and v2 is a nonunit in GR(p2 , m). Then we have v1 xq1 is a unit and v2 xq2 is a nonunit in GR(p2 , m)[x]/xn − 1 by Lemmas 3.3 and 3.4. So, v is a unit in GR(p2 , m)[x]/xn − 1; this is because, if not, since GR(p2 , m)[x]/xn − 1 is a local ring, v1 xq1 = v − v2 xq2 is a nonunit in GR(p2 , m)[x]/xn − 1, a contradiction. But, v cannot be a unit in GR(p2 , m)[x]/xn − 1. Case 2. Suppose that both v1 and v2 are units in GR(p2 , m). Then, by reducing v modulo p, we have a nonzero codeword v in the residue code Res(C) associated with an ideal (x − 1)r  of R[x]/xn − 1. Then Res(C) has the minimum Hamming weight less than or equal to 2 as wtH (v) = 2. However, by Proposition 2.4, we get  4 if p = 2, dH (Res(C)) = dH ((x − 1)r ) ≥ p if p ≥ 3, since 2k−1 < r ≤ n − 1 if p = 2 and (p − 1)pk−1 ≤ r ≤ n − 1 if p ≥ 3 by Case 2 of Proposition 3.1, where dH (Res(C)) is the minimum Hamming weight of Res(C); this is impossible since the minimum Hamming weight of Res(C) is less than or equal to 2. From Case 1 and Case 2, we find that both v1 and v2 are nonunits in GR(p2 , m) so that wtL (v) ≥ 2p as wtL (vi xqi ) ≥ p for i = 1, 2. Next, we want to show that there is a codeword v in C such that wtL ( v ) = 2p. By Proposition 2.1 (ii) and Lemma 3.3, we have v1 xq1 + v2 xq2 = (t1 p)xq1 + (t2 p)xq2 , q1 q1  q2 q2  i i = (t1 + t2 )p + (t1 p) i=1 i=1 i (x − 1) i (x − 1) + (t2 p) (3)

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with a nonzero ti ∈ R (i = 1, 2). Moreover, there is a polynomial hi (x) = n−1 j 2 n j=0 hij (x − 1) ∈ GR(p , m)[x]/x − 1 (i = 1, 2) such that 

 ai (x − 1)i h1 (x) + p(x − 1)n−r h2 (x) q1 q1  q2 q2  i i = (t1 + t2 )p + (t1 p) i=1 i=1 i (x − 1) . i (x − 1) + (t2 p)

(x − 1)r + p

n−r−1 i=0

(4)

We claim that t1 + t2 is contained in p in GR(p2 , m). By substitution x = 1 in Eq. (4), we have pa0 h10 = (t1 + t2 )p in GR(p2 , m). Then we get a0 h10 − (t1 + t2 ) ∈ p

(5)

in GR(p2 , m). Moreover, by reducing Eq. (4) modulo p, we obtain (x − 1)r h1 (x) = 0

(6)

in R[x]/xn −1. So, we get h10 ∈ p in GR(p2 , m); this is because, we note that (x − 1)r h1 (x) = h10 (x − 1)r + h11 (x − 1)r+1 + · · · + h1(n−r−1) (x − 1)n−1 in R[x]/xn − 1 by Lemma 3.3 (ii). By Eq. (6), we then have h1j ∈ p of GR(p2 , m) for 0 ≤ j ≤ n − r − 1. Thus, our claim is proved by Eq. (5). Then we have (t1 p)xq1 + (t2 p)xq2 = (t1 p)xq1 − (t1 p)xq2 ∈ C since t1 + t2 ∈ p of GR(p2 , m). And we obtain q1 q2 q1 q2 t−1 ∈C 1 ((t1 p)x − (t1 p)x ) = px − px

since t1 is a unit in GR(p2 , m) by Proposition 2.1 (ii). It gives that there is a codeword v = pxq1 − pxq2 in C such that wtH ( v ) = 2 and wtL ( v ) = 2p. 2 Lemma 3.6. Let C be a cyclic self-dual code over GR(p2 , m) of length n = pk with (p −1)pk−1 < r ≤ n −1. Let v be a codeword with wtH (v) > 2. Then we have wtL (v) ≥ 2p. s Proof. Let v = i=0 vi xqi be a codeword in C with vi = 0 for all 0 ≤ i ≤ s. Assume that vi is a nonunit in GR(p2 , m) for all 0 ≤ i ≤ s. Then we get wtL (v) > 2p since wtH (v) > 2 by our assumption. If not, that is, there exists a unit vi in GR(p2 , m) for 0 ≤ i ≤ s. Then, by reducing v modulo p, we obtain v ∈ R[x]/xn − 1 and v is a nonzero codeword in the residue code Res(C) associated with an ideal (x − 1)r  of R[x]/xn − 1. By Proposition 2.4, we have wtH (v) ≥ 2p since r > (p − 1)pk−1 , so we get 2p ≤ wtH (v) ≤ wtH (v) ≤ wtL (v). 2

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Let C be a cyclic self-dual code over GR(p2 , m) of length n = pk with the residue degree r. If r = n, then we assume that any nonzero codeword in C has the Hamming weight at least 2 since C has the minimum Hamming weight 2 by Proposition 3.2. Moreover, if r = (p − 1)pk−1 and p = 2, then C is associated with a principal n−r−1 ideal (x − 1)r + pa0 + p i=1 ai (x − 1)i  of GR(p2 , m)[x]/xn − 1 with ai ∈ R (0 ≤ i ≤ n − r − 1) and a0 = 0, 1 by Proposition 3.1, Case 2 (i). We need to prove the following lemmas for proving Theorem 1.1, Case 3 (ii). Lemma 3.7. Let C be a cyclic self-dual code over GR(p2 , m) of length n = pk with e e m−1 r = (p − 1)pk−1 , where p is an odd prime. Let v = j=0 vj xqj = j=0 ( =0 vj ξ  )xqj be a nonzero codeword in C with vj = 0 (0 ≤ j ≤ e), 0 ≤ q0 < · · · < qe ≤ n − 1 and e ≥ 1. Then v satisfies the following properties: e (i) j=0 vj = pa0 v0 = pa0 v1 = · · · = pa0 ve (ii) v0 ≡ v1 ≡ · · · ≡ ve (mod p) in GR(p2 , m) (iii) if every vj is a unit in GR(p2 , m) for 0 ≤ j ≤ e, then we have p | wtH (v). In addition, if wtL (v) < 2p, then v also satisfies the following properties: (iv) every vj is a unit in GR(p2 , m) for 0 ≤ j ≤ e. (v) wtH (v) = e + 1 is equal to p. (vi) (v0 , . . . , ve ) = (v0 ξ  , . . . , ve ξ  ) for any  with 0 ≤  ≤ m − 1. Proof. (i) Let s be an integer with 0 ≤ s ≤ e. We note that there is a codeword v  = xτ v =

e 



vj xqj ∈ C

j=0

such that ve = vs and qe  = r with 0 ≤ q0  < · · · < qe  ≤ n − 1 for some non-negative integer τ since C is a cyclic code. By Lemma 3.3, we have 

v =

e 

q0   q0 

vj

+

v0

j=0

j=1

j

(x − 1) + · · · + vs j

r  r j=1

j

(x − 1)j .

(7)

Moreover, by Eq. (7), we get e j=0

 q0   q0   r   (x − 1)j + · · · + vs j=1 jr (x − 1)j j=1 j   n−r−1 = (x − 1)r + pa0 + p i=1 ai (x − 1)i h(x)

vj + v0

for some nonzero polynomial h(x) = Then we can say that

n−1 i=0

(8)

hi (x − 1)i ∈ GR(p2 , m)[x]/xn − 1.

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B. Kim, Y. Lee / Finite Fields and Their Applications 45 (2017) 107–130

h0 = vs in R,

(9)

where x denotes the reduction of x ∈ GR(p2 , m) (mod p); this is because, by reducing Eq. (8) modulo p, we have v  = (x − 1)r h(x)

(10)

in R[x]/xn − 1. Especially, the right hand side of Eq. (10) is (x − 1)r h(x) = h0 (x − 1)r + h1 (x − 1)r+1 + · · · + hn−r−1 (x − 1)n−1 by Lemma 3.3 (ii). So, we obtain h0 = vs and h1 = · · · = hn−r−1 = 0 in R since the degree of the left hand side of Eq. (10) is equal to r. e By substitution x = 1 in Eq. (8), we get j=0 vj = pa0 h0 in GR(p2 , m). Thus, we  e e have j=0 vj = j=0 vj = pa0 h0 = pa0 vs for any 0 ≤ s ≤ e by Eq. (9); the result follows. (ii) By (i), without loss of generality, we get pa0 v0 = pa0 v1 in GR(p2 , m). Then we have pa0 (v0 − v1 ) = 0 in GR(p2 , m), so a0 (v0 − v1 ) ∈ p. This implies that v0 − v1 ∈ p since a0 is a unit in GR(p2 , m); so we get the results. (iii) First, wtH (v) is equal to e + 1 since vj is a nonzero in GR(p2 , m) for all j. We claim that e + 1 ≡ 0 (mod p). Without loss of generality, we get e 

vj ≡ (e + 1)v0

(mod p)

j=0

in GR(p2 , m) by (ii). Moreover, by (i), we have (e + 1)v0 ≡ 0 (mod p) in GR(p2 , m). Then we have e + 1 ≡ 0 (mod p) since v0 is a unit in GR(p2 , m). Thus, the claim is proved. (iv) Assume not, that is, there exist some nonunits vj in GR(p2 , m) with 0 ≤ j ≤ e. Then, by (ii), every vj is a nonunit in GR(p2 , m) for 0 ≤ j ≤ e. Since wtH (v) = e + 1 ≥ 2, we get wtL (v) ≥ wtL (v0 ) + wtL (v1 ) ≥ 2p as wtL (vj ) ≥ p with j = 1, 2. But, it is impossible by our assumption. (v) By (iii) and (iv), we have p | wtH (v). We note that wtH (v) ≤ wtL (v) < 2p, so the result follows. (vi) If one of v0 , v1 , · · · , ve (0 ≤  ≤ m − 1) is nonzero in Zp2 , then we first claim that v0 , v1 , · · · , ve are all nonzeros. Assume not, that is, vi = 0 and vj = 0 for some 0 ≤ i, j ≤ e with j = i. Let v0 = v00 ξ 0 + v01 ξ 1 + · · · + v0m−1 ξ m−1 and v1 = v10 ξ 0 + v11 ξ 1 + · · · + v1m−1 ξ m−1 . Without loss of generality, suppose that v00 = 0 and v10 = 0 in Zp2 . By (i), we get

B. Kim, Y. Lee / Finite Fields and Their Applications 45 (2017) 107–130

pa0

m−1 

v0 ξ  = pa0

=0

m−1 

119

v1 ξ 

=0

m−1 in GR(p2 , m). So, we have pa0 =0 (v0 − v1 )ξ  = 0 in GR(p2 , m). Since a0 is a  m−1 unit in GR(p2 , m), =0 (v0 −v1 )ξ  should be an element in the maximal ideal p 2 of GR(p , m). That is, we obtain v0 − v1 ≡ 0 (mod p) in Zp2 for all 0 ≤  ≤ m − 1. It gives that v00 − v10 = −v10 is divisible by p in Zp2 . Then there is v1s which is a unit in Zp2 for some 0 < s ≤ m − 1; this is because, if v1 is a nonunit in Zp2 for all 0 ≤  ≤ m − 1, then v1 is also nonunit in GR(p2 , m). But it is impossible since v1 is a unit by (iv). Thus, we have L (v10 ) + wt L (v1s ) ≥ p + 1. wtL (v1 ) ≥ wtL (v10 ξ 0 ) + wtL (v1s ξ s ) = wt Clearly, we get

wtL (v) =

e 

wtL (vj ) ≥ p + 1 +

j=0

e 

wtL (vj ) ≥ p + 1 + e = 2p

j≥0,j=1

since e + 1 = p by (v). It is a contradiction since we assume that wtL (v) < 2p. Hence v0 , · · · , ve are either all zeros or all nonzeros in Zp2 for each 0 ≤  ≤ m − 1. It remains to show that there exists a unique  with 1 ≤  ≤ m −1 for each vj = vj ξ  m−1 (0 ≤ j ≤ e). Let I = {i | v0i = 0, where v0 = i=0 v0i ξ i } and |I| the cardinality of the set I. To show |I| = 1, suppose that |I| ≥ 2 for contradiction. Then, without loss of generality, we let (v0 , . . . , ve ) = (v00 ξ 0 + · · · + v0τ ξ τ , . . . , ve0 ξ 0 + · · · + veτ ξ τ ) with τ ≥ 2 and v0i = 0 for all 0 ≤ i ≤ τ by the first claim. So, we get wtL (v) ≥ τ p ≥ 2p by (v), but it is contradictory to our assumption. Thus, |I| = 1. 2 Remark 3.8. Let C be a cyclic self-dual code over GR(p2 , m) of length n = pk with an odd prime p and r = (p − 1)pk−1 . In the following Lemmas 3.9 and 3.10, if wtL (v) < 2p p−1 for a codeword v ∈ C, we can assume that the codeword v has the form i=0 vi xqi , where vi is a unit in Zp2 with 0 ≤ q0 < . . . < qp−1 ≤ n − 1 for all 0 ≤ i ≤ p − 1. This p−1 follows from that v = i=0 (vi ξ  )xqi with  = 0 by Lemma 3.7 (iv), (v) and (vi). Lemma 3.9. Let C be a cyclic self-dual code over GR(p2 , m) of length n = pk with r = (p − 1)pk−1 , where p is an odd prime. Then we have wtL (C) ≥ 2p − 1. Furthermore, p−1 wtL (C) = 2p − 1 if and only if there is a codeword v = i=0 vi xqi ∈ C such that  (v0 , . . . , vp−1 ) =

±(4, 1, 1) or ± (7, 7, 1)

if p = 3,

±((ps + 1), 1, . . . , 1)

if p ≥ 5,

with 0 < s < p − 1 (up to cyclic shift).

(11)

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t−1 Proof. We note that C is associated with a principal ideal (x − 1)r + pa0 + p i=1 ai (x − 1)i  ∈ GR(p2 , m)[x]/xn − 1 with ai ∈ GR(p2 , m)/p (0 ≤ i ≤ t − 1), a0 = 0, 1 and t = n − r = pk−1 since r = (p − 1)pk−1 . Suppose that there exists a codeword v in C p−1 such that wtL (v) < 2p. Then a codeword v has the form i=0 vi xqi , where vi is a unit in Zp2 (0 ≤ i ≤ p − 1) and 0 ≤ q0 < . . . < qp−1 ≤ n − 1 by Remark 3.8. Moreover, by Lemma 3.7 (ii), we obtain vi ≡ vj (mod p) in GR(p2 , m) for all 0 ≤ i, j ≤ p − 1. Thus, we have vi ≡ ±1 (mod p) for all 0 ≤ i ≤ p −1; if not, we have wtL (v) ≥ 2p, contradiction L (−vi ) for all L (vi ) = wt to our assumption. We may assume that vi ≡ 1 (mod p) as wt 0 ≤ i ≤ p − 1. Let vi = psi + 1 (0 ≤ si ≤ p − 1) and nv = |{0 ≤ i ≤ p − 1 | si = 0}| for any 0 ≤ i ≤ p − 1. (i) If nv = 0, then we get (v0 , . . . , vp−1 ) = (1, . . . , 1) and wtL (v) = p. Thus, p−1 2 i=0 vi = p = pa0 by Lemma 3.7 (i), where a0 ∈ GR(p , m)/p; so, a0 = 1, which is impossible. (ii) Suppose that nv = 1; without loss of generality, assume s0 = 0. Moreover, we note p−1 that s0 = p − 1; if not, v = (p(p − 1) + 1, 1, . . . , 1), so i=0 vi = p2 = 0 = pa0 , L (ps0 + 1) = p; therefore, then we have a0 = 0, contradiction. As s0 = 0, p − 1, wt wtL (v) = 2p − 1 (in this case, (v0 , . . . , vp−1 ) = (ps0 + 1, 1, . . . , 1) with 1 ≤ s0 ≤ p − 2 (up to cyclic shift)). (iii) Suppose that nv ≥ 2. If nv ≥ 3, then clearly we have wtL (v) ≥ 3(p − 1) + p − 3 = 4p −6 ≥ 2p for any odd prime. Now, if nv = 2, assume that 0 < si , sj ≤ p −1 (i = j). Then we get wtL (v) ≥ 2(p − 1) + p − 2 = 3p − 4; in particular, the equality holds if si = sj = p − 1. So, we have wtL (v) ≥ 3p − 4 = 5 = 2p − 1 if p = 3 and wtL (v) > 2p if p ≥ 5 (in this case, if p = 3, wtL (v) = 2p − 1 = 5 if and only if (v0 , v1 , v2 ) = (p(p − 1) + 1, p(p − 1) + 1, 1) = (7, 7, 1) (up to cyclic shift)). By (i), (ii) and (iii), we obtain that wtL (C) ≥ 2p − 1. Furthermore, if p ≥ 5, then wtL (v) = 2p − 1 happens exactly when nv = 1 and (v0 , . . . , vp−1 ) = (ps0 +1, 1, . . . , 1) with 0 < s0 < p−1. Now, if p = 3, then wtL (v) = 2p−1 if and only if nv = 1 with (v0 , v1 , v2 ) = (ps0 + 1, 1, 1) = (4, 1, 1) or nv = 2 with (v0 , v1 , v2 ) = (7, 7, 1) (up to cyclic shift). Hence, the result follows. 2 The following lemma shows exactly when there exists a codeword of Lee weight 2p −1. Lemma 3.10. Let C be a cyclic self-dual code over GR(p2 , m) of length n = pk with the residue degree r = (p −1)pk−1 (then C is associated with a principal ideal (x −1)r +pa0 + t−1 p i=1 ai (x − 1)i  ∈ GR(p2 , m)[x]/xn − 1 with ai ∈ GR(p2 , m)/p (0 ≤ i ≤ t − 1), a0 = 0, 1 and t = n − r = pk−1 ). Then I is exactly in the following form: I = (x − 1)r + pa0 

(12)

B. Kim, Y. Lee / Finite Fields and Their Applications 45 (2017) 107–130

p−1

if and only if there exists a codeword v =

i=0

vi xqi in C such that

 (v0 , . . . , vp−1 ) =

121

±(4, 1, 1) or ± (7, 7, 1)

if p = 3,

±((ps + 1), 1, . . . , 1)

if p ≥ 5,

(13)

with 0 < s < p − 1 (up to cyclic shift). Proof. We note that p−1 (p−1)pk−1

(x − 1)

=

i=0 (pκi

p−1

i=0 (pκi

+ 1)xi + 1)x

if k = 1,

i·pk−1

pk−1 (p−2)

+ p(x − 1)

g(x)

if k ≥ 2,

(14)

for some nonzero polynomial g(x) ∈ (GR(p2 , m)/p)[x]/xn − 1 with 0 ≤ κi ≤ p − 1;   for k = 1, we can prove (−1)p−1−i p−1 ≡ 1 (mod p) for all 0 ≤ i ≤ p − 1 since i p−1 p−1 p = i for all 0 ≤ i ≤ p − 1. Moreover, by replacing x by xp in (x − 1)p−1 = i−1 + i  p−1 i i=0 (pκi + 1)x , we obtain the polynomial in Eq. (14) for k = 2. For k ≥ 3, it can be proved by mathematical induction. From Eq. (14), we have (x − 1)(p−1)p

k−1

=

p−1 

xi·p

k−1

(mod p).

(15)

i=0

Furthermore, we note that k−1

p(x − 1)p

k−1

= −p + pxp

(16)

in GR(p2 , m)[x]/xn − 1; this can be shown by mathematical induction. Moreover, k−1 p(x − 1)p is a codeword in C since the torsion degree t of C is equal to pk−1 . Then, by Eq. (14), we have a codeword: p−1  k−1 (pκi + 1)xi·p = i=0

 (x − 1)r

if k = 1, pk−1 (p−2)

(x − 1) − p(x − 1) r

g(x)

if k ≥ 2,

(17)

in C. p−1 Our first claim is that if there is a codeword v = i=0 vi xqi such that (v0 , . . . , vp−1 ) = ±((sp + 1), 1, . . . , 1) with 0 ≤ q0 < . . . < qp−1 ≤ n − 1 and 1 ≤ s ≤ p − 2, then we have qi = i · pk−1 and s ≡ a0 − 1

(mod p).

First, we consider (v0 , . . . , vp−1 ) = ((sp + 1), 1, . . . , 1) case. Let v = (sp + 1)xq0 + xq1 + · · · + xqp−1 be a codeword in C with 0 ≤ q0 < · · · < qp−1 ≤ n − 1 (without loss of generality, we assume the ordering of qi ’s). By a proper cyclic shift of v by xτ with some integer τ (0 ≤ τ ≤ n − 1), we get a codeword

B. Kim, Y. Lee / Finite Fields and Their Applications 45 (2017) 107–130

122





v  = xτ · v = (sp + 1)xq0 + xq1 + · · · + xr  in C with 0 ≤ q0 < · · · < qp−1 = r = (p − 1)pk−1 . Moreover, we note that





(sp + 1)xq0 + xq1 + · · · + xr = ((x − 1)r + pa0 + p

t−1 

ai (x − 1)i )h(x)

(18)

i=1

for some nonzero polynomial h(x) = reducing Eq. (18) modulo p, we get 

n−1 i=0

hi (x − 1)i of GR(p2 , m)[x]/xn − 1. By



xq0 + xq1 + · · · + xr = (x − 1)r h(x) = h0 (x − 1)r + h1 (x − 1)r+1 + · · · + hn−r−1 (x − 1)n−1

(19)

in (GR(p2 , m)/p)[x]/xn − 1; the second equality is from Lemma 3.3 (ii). Moreover, we get h0 = 1 and h1 = · · · = hn−r−1 = 0; this is because, the left hand side of q0 q0  q1 q1  r r i i i Eq. (19) can be written by i=0 i=0 i (x − 1) + i=0 i (x − 1) in i (x − 1) + 2 n (GR(p , m)/p)[x]/x − 1 by Lemma 3.3 (ii). So, by Eq. (15), we obtain  q0 = 0, q1 = pk−1 , . . . , qp−1 = qr = (p − 1)pk−1 .

Moreover, by substitution x = 1 in Eq. (18), we have p(s +1) = pa0 h0 = pa0 since h0 = 1. It means that s ≡ a0 − 1 (mod p) in GR(p2 , m). If (v0 , . . . , vp−1 ) = −((sp + 1), 1, . . . , 1), then we have the same result by similar process as above. Thus, the first claim follows. From the previous claim, if p = 3, then it follows that (v0 , v1 , v2 ) = ±(4, 1, 1) is contained in C if and only if (v0 , v1 , v2 ) = ±(7, 7, 1) belongs to C. In detail, if there is a 2 codeword v = i=0 vi xqi in C such that (v0 , v1 , v2 ) = ±(7, 7, 1) with 0 ≤ q0 < q1 < q2 ≤ n − 1, then we have q0 = 0, q1 = 3k−1 and q2 = r by using the same reasoning as above k−1 k−1 k−2 k−2 k−1 k−2 k−2 and (x − 1)2·3 = x2·3 + 3x5·3 + 6x4·3 + 7x3 + 6x2·3 + 3x3 + 1. It 2 qi follows that there is a codeword v = i=0 vi x in C with (v0 , v1 , v2 ) = ±(4, 1, 1) if and 2  only if there exists a codeword v = i=0 vi xqi in C with (v0 , v1 , v2 ) = ±(7, 7, 1) since q0 = q0 = 0, q1 = q1 = 3k−1 , q2 = q2 = r and vi = −2vi for 0 ≤ i ≤ 2 (up to cyclic shift). p−1 Thus, it remains to show that there exists a codeword v = i=0 vi xqi ∈ C such that (v0 , · · · , vp−1 ) = ±((ps + 1), 1, . . . , 1)

(20)

with 0 ≤ q0 < . . . < qp−1 ≤ n − 1 and 1 ≤ s ≤ p − 2 (up to cyclic shift) for an odd prime if and only if I is exactly of the form in Eq. (12). Suppose that I = (x − 1)r + pa0  of GR(p2 , m)[x]/xn − 1. Then, by Eq. (17), we get a codeword p−1  i=0

k−1

(pκi + 1)xi·p

B. Kim, Y. Lee / Finite Fields and Their Applications 45 (2017) 107–130

123

in C. Furthermore, p−1 k−1 pa0 + i=0 (pκi + 1)xi·p p−1 p−1 k−1 k−1 = i=0 xi·p + p i=0 κi xi·p + pa0 p−1 i·pk−1 p−1 i = i=0 x + i=0 κi (p(x − 1)pk−1 + p) + pa0 p−1 p−1 p−1 k−1 k−1 = i=0 xi·p + p i=0 κi (xi − 1)p + p i=0 κi + pa0 p−1 p−2 k−1  pk−1 + p p−2 κi + pa0 = i=0 xi·p + p i=1 κi ((x − 1)g(x)) i=1 p−1 i·pk−1 p−2  pk−1 = i=1 x + p(a0 + κ1 + . . . + κp−2 ) + 1 + p i=1 κi ((x − 1)g(x)) p−1 p−2 k−1  pk−1 , = i=1 xi·p + p(a0 − 1) + 1 + p i=1 κi ((x − 1)g(x))

(21)

 for some nonzero polynomial g(x)  in (GR(p2 , m)/p)[x]/ where (xi − 1) = (x − 1)g(x) xn − 1 with i ≥ 1; the fourth equation is from κ0 = κp−1 = 0 and the sixth equation p−2 is from p( i=1 κi + 1) = 0 in GR(p2 , m) by substitution x = 1 in Eq. (14). Moreover, from Eq. (21), we obtain a codeword p−1 

xi·p

k−1

+ p(a0 − 1) + 1 ∈ C;

(22)

i=1

p−2  pk−1 ∈ C since the torsion degree t = this is because, we have p i=1 κi ((x − 1)g(x)) n − r = pk−1 . Let s = a0 − 1. Then we say that s = 0 and s = p − 1 in GR(p2 , m); if s = 0, then we get a codeword p−1 

xi + 1 ∈ C

i=1

p−1 by Eq. (22). Then, by Lemma 3.7 (i), we have i=0 1 = p = pa0 , so a0 = 1 since 2 a0 ∈ GR(p , m)/p. But, it’s a contradiction since a0 = 0, 1. Similarly, if s = p − 1, it follows that a0 = 0 by the same reason as above. Hence, we get 1 ≤ s ≤ p − 2. It means that there is a codeword v ∈ C which satisfies the condition of Eq. (20). To prove the converse statement, we assume that there exists a codeword v = p−1 qi i=0 vi x in C such that (v0 , . . . , vp−1 ) = ((ps +1), 1, . . . , 1) with 0 ≤ q0 < . . . < qp−1 ≤ n − 1 and 1 ≤ s ≤ p − 2 by Remark 3.8. Then, by the first claim, there exists a codeword v = (ps + 1) +

p−1 

xi·p

k−1

(23)

i=1

with s = a0 − 1 in C (as we mentioned above, a0 − 1 = 0 and a0 − 1 = 1). So, we obtain a codeword in C ((x − 1) + pa0 + p r

t−1  i=1

ai (x − 1)i ) − v

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B. Kim, Y. Lee / Finite Fields and Their Applications 45 (2017) 107–130

 t−1 p−2  pk−1 ) p( i=1 ai (x − 1)i + i=1 κi ((x − 1)g(x)) = p−2 t−1  pk−1 + (x − 1)pk−1 (p−2) g(x)) p( i=1 ai (x − 1)i + i=1 κi ((x − 1)g(x))

if k = 1, if k ≥ 2,

t−1 by Eq. (17) and (21). Thus, we have a codeword p i=1 ai (x − 1)i in C; this is because, p−2  pk−1 and p(x − 1)pk−1 (p−2) g(x) are codewords in C since pk−1 both p i=1 κi ((x − 1)g(x)) is the torsion degree t of C. Then we have a1 = a2 = . . . = at−1 = 0; if not, it’s a contradiction with the fact that the torsion degree t = pk−1 . Hence, we have I = (x − 1)r + pa0  of GR(p2 , m)[x]/xn − 1. Similarly, we have the same result if (v0 , . . . , vp−1 ) = −((ps + 1), 1, . . . , 1). Hence, the result follows. 2 Now, we are ready to prove Theorem 1.1. Proof of Theorem 1.1. • Case 1. If r = n, then we have I = p of R[x]/xn − 1 by Proposition 3.1, Case 1 (i). Then, for any nonzero codeword w = (w1 , . . . , wn ) ∈ C, every wi is a nonunit in GR(p2 , m). Thus, we have wtL (w) ≥ p. Furthermore, the minimum Lee weight of C is equal to p; this is because, the minimum Hamming weight of C is equal to 1 and there is a codeword (0, . . . , 0, p, 0, . . . , 0) in C which gives the minimum Lee weight of C. 2 • Case 2 and Case 3 (i). In this case, we assume that r = n and r = (p − 1)pk−1 (that is, (p − 1)pk−1 < r < n). First of all, if wtH (w) > 2 for w ∈ C, then wtL (w) ≥ 2p by Lemma 3.6. By Propositions 3.2 and 3.5, if r = n, then C has a codeword v whose Hamming weight 2 and its Lee weight is greater than or equal to 2p. Furthermore, there exists a codeword v in C whose Lee weight is exactly the same as 2p. Therefore, the minimum Lee weight of C is equal to 2p. Thus, Case 3 (i) is proved. In particular, for p = 2 and r = n, if C is self-dual, then we have 2k−1 < r ≤ n − 1 by Case 2 of Proposition 3.1. Thus, Case 2 is also proved by the same reasoning as above. 2 • Case 3 (ii). For r = (p − 1)pk−1 and p = 2, it follows that 2p − 1 ≤ wtL (C) ≤ 2p from Lemmas 3.5 and 3.9. Moreover, wtL (C) = 2p − 1 if and only if there is a codeword v which has the form in Eq. (11). Lemma 3.10 shows that the existence of such codeword v is equivalent to that f (x) = a0 . The result thus follows. 2 4. Implementation: cyclic self-dual codes over GR(22 , 1), GR(32 , 1) and GR(32 , 2) An extremal code C means that its minimum weight meets the largest of the applicable bound. According to Theorem 1.1, most of cyclic self-dual codes over Galois rings we found are extremal with respect to the Lee weights.

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4.1. Cyclic self-dual codes over GR(22 , 1) of lengths 16 and 32 In [1], the authors find the generators of cyclic self-dual codes over GR(22 , 1) ∼ = Z4 of length 8. We find all the ideals which are associated with cyclic self-dual codes over Z4 of lengths 16 and 32 and their Lee weight enumerators by using Proposition 3.1 and Magma. Theorem 4.1. We find all the ideals of Z4 [x]/xn − 1 which are associated with cyclic self-dual codes over Z4 of lengths n = 16 and 32 and we also find their Lee weight enumerators. • n = 16 There are exactly 32 principal ideals and 27 non-principal ideals of Z4 [x]/xn − 1 which are associated with cyclic self-dual codes over Z4 of length n = 16. Among them, there are 58 extremal cyclic self-dual codes over Z4 of length 16 with respect to Lee weights. Moreover, there are exactly 19 Lee weight enumerators for cyclic self-dual codes over Z4 of length 16 in Table 1. • n = 32 There are exactly 512 principal ideals and 507 non-principal ideals of Z4 [x]/xn − 1 which are associated with cyclic self-dual codes over Z4 of length n = 32. Among them, there are 1018 extremal cyclic self-dual codes over Z4 of length 32 with respect to Lee weights. Moreover, there are exactly 168 Lee weight enumerators for cyclic self-dual codes over Z4 of length 32 in Table 2. Tables 1 and 2 are not entire lists of ideals. A complete list of the ideals can be found in [11]. 4.2. Cyclic self-dual codes over GR(32 , 1) of lengths up to 27 We obtain the following implementation result using Proposition 3.1. Theorem 4.2. We find all the ideals of Z9 [x]/xn − 1 which are associated with cyclic self-dual codes over Z9 of lengths n = 3, 9 and 27. • n=3 There are exactly 2 principal ideals of Z9 [x]/x3 − 1 which are associated with cyclic self-dual codes over Z9 of length 3. • n=9 There are exactly 6 principal ideals and 2 non-principal ideals of Z9 [x]/x9 −1 which are associated with cyclic self-dual codes over Z9 of length 9. Among then, there are 6 extremal cyclic self-dual codes over Z9 of length 9 with respect to Lee weights.

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Table 1 Cyclic self-dual codes over Z4 of length 16. No. Ideals 1 2

Lee weight enumerators X 32 + 16X 30 Y 2 + 120X 28 Y 4 + 560X 26 Y 6 1820X 24 Y 8 + 4368X 22 Y 10 + 8008X 20 Y 12 11440X 18 Y 14 + 12870X 16 Y 16 + 11440X 14 Y 18 8008X 12 Y 20 + 4368X 10 Y 22 + 1820X 8 Y 24 560X 6 Y 26 + 120X 4 Y 28 + 16X 2 Y 30 + Y 32

+ + + +

2

(x − 1)15 + 2

X 32 + 120X 28 Y 4 + 1820X 24 Y 8 + 8008X 20 Y 12 + 45638X 16 Y 16 + 8008X 12 Y 20 + 1820X 8 Y 24 + 120X 4 Y 28 + Y 32

3

(x − 1)13 , 2(x − 1)3 

X 32 + 24X 28 Y 4 + 860X 24 Y 8 + 12712X 20 Y 12 + 38342X 16 Y 16 + 12712X 12 Y 20 + 860X 8 Y 24 + 24X 4 Y 28 + Y 32

4

(x − 1)14 + 2

X 32 + 56X 28 Y 4 + 1180X 24 Y 8 + 11144X 20 Y 12 + 40774X 16 Y 16 + 11144X 12 Y 20 + 1180X 8 Y 24 + 56X 4 Y 28 + Y 32

5 6 7

(x − 1)14 , 2(x − 1)2  (x − 1)12 + 2 (x − 1)12 , 2(x − 1)4 

8

(x − 1)14 + 2 + 2(x − 1)

9

(x − 1)14 + 2(x − 1), 2(x − 1)2 

10

(x − 1)12 + 2 + 2(x − 1)2 

11

(x − 1)12 + 2(x − 1)2 , 2(x − 1)4 

12

(x − 1)12 + 2 + 2(x − 1)3 

13 14 15

(x − 1)12 + 2(x − 1)3 , 2(x − 1)4  (x − 1)13 + 2(x − 1)2 , 2(x − 1)3  (x −1)13 +2(x −1) +2(x −1)2 , 2(x −1)3 

16

(x − 1)12 + 2 + 2(x − 1)2 + 2(x − 1)3 

17 18 19

(x−1)12 +2(x−1)2 +2(x−1)3 , 2(x−1)4  (x − 1)13 , 2(x − 1)3  (x − 1)13 + 2(x − 1), 2(x − 1)3 

20

(x − 1)10 + 2

21

(x − 1)10 + 2 + 2(x − 1)2 

22

(x − 1)10 + 2 + 2(x − 1)4 

23

(x − 1)10 + 2 + 2(x − 1)2 + 2(x − 1)4 

X 32 + 56X 28 Y 4 + 924X 24 Y 8 + 2048X 22 Y 10 + 3976X 20 Y 12 + 14336X 18 Y 14 + 22854X 16 Y 16 + 14336X 14 Y 18 + 3976X 12 Y 20 + 2048X 10 Y 22 + 924X 8 Y 24 + 56X 4 Y 28 + Y 32 X 32 + 24X 28 Y 4 + 128X 26 Y 6 + 348X 24 Y 8 1664X 22 Y 10 + 6568X 20 Y 12 + 14592X 18 Y 14 18886X 16 Y 16 + 14592X 14 Y 18 + 6568X 12 Y 20 1664X 10 Y 22 + 348X 8 Y 24 + 128X 6 Y 26 + 24X 4 Y 28 Y 32

+ + + +

X 32 + 24X 28 Y 4 + 604X 24 Y 8 + 2048X 22 Y 10 + 5544X 20 Y 12 + 14336X 18 Y 14 + 20422X 16 Y 16 + 14336X 14 Y 18 + 5544X 12 Y 20 + 2048X 10 Y 22 + 604X 8 Y 24 + 24X 4 Y 28 + Y 32

X 32 + 24X 28 Y 4 + 860X 24 Y 8 + 12712X 20 Y 12 + 38342X 16 Y 16 + 12712X 12 Y 20 + 860X 8 Y 24 + 24X 4 Y 28 + Y 32

X 32 + 40X 28 Y 4 + 64X 26 Y 6 + 700X 24 Y 8 + 1344X 22 Y 10 + 7064X 20 Y 12 + 10880X 18 Y 14 + 25350X 16 Y 16 + 10880X 14 Y 18 + 7064X 12 Y 20 + 1344X 10 Y 22 +700X 8 Y 24 +64X 6 Y 26 +40X 4 Y 28 +Y 32 X 32 +8X 28 Y 4 +192X 26 Y 6 +124X 24 Y 8 +960X 22 Y 10 + 9656X 20 Y 12 + 11136X 18 Y 14 + 21382X 16 Y 16 + 11136X 14 Y 18 + 9656X 12 Y 20 + 960X 10 Y 22 + 124X 8 Y 24 + 192X 6 Y 26 + 8X 4 Y 28 + Y 32

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Table 2 Cyclic self-dual codes over Z4 of length 32. No. Ideals Lee weight enumerators 1 2 X 64 + 32X 62 Y 2 + 496X 60 Y 4 + 4960X 58 Y 6 + 35960X 56 Y 8 + 201376X 54 Y 10 + 906192X 52 Y 12 + 3365856X 50 Y 14 + 10518300X 48 Y 16 + 28048800X 46 Y 18 + 64512240X 44 Y 20 + 129024480X 42 Y 22 + 225792840X 40 Y 24 + 347373600X 38 Y 26 + 471435600X 36 Y 28 + 565722720X 34 Y 30 + 601080390X 32 Y 32 + 565722720X 30 Y 34 + 471435600X 28 Y 36 + 347373600X 26 Y 38 + 225792840X 24 Y 40 + 129024480X 22 Y 42 + 64512240X 20 Y 44 + 28048800X 18 Y 46 + 10518300X 16 Y 48 + 3365856X 14 Y 50 + 906192X 12 Y 52 + 201376X 10 Y 54 + 35960X 8 Y 56 + 4960X 6 Y 58 + 496X 4 Y 60 + 32X 2 Y 62 + Y 64 2

(x + 1)31 + 2

X 64 + 496X 60 Y 4 + 35960X 56 Y 8 + 906192X 52 Y 12 + 10518300X 48 Y 16 + 64512240X 44 Y 20 + 225792840X 40 Y 24 + 471435600X 36 Y 28 + 2748564038X 32 Y 32 + 471435600X 28 Y 36 + 225792840X 24 Y 40 + 64512240X 20 Y 44 + 10518300X 16 Y 48 + 906192X 12 Y 52 + 35960X 8 Y 56 + 496X 4 Y 60 + Y 64

3

(x + 1)30 + 2

X 64 + 240X 60 Y 4 + 18040X 56 Y 8 + 452816X 52 Y 12 + 5325596X 48 Y 16 + 40118256X 44 Y 20 + 232175944X 40 Y 24 + 760524368X 36 Y 28 + 2217736774X 32 Y 32 + 760524368X 28 Y 36 + 232175944X 24 Y 40 + 40118256X 20 Y 44 + 5325596X 16 Y 48 + 452816X 12 Y 52 + 18040X 8 Y 56 + 240X 4 Y 60 + Y 64

4

(x + 1)30 + 2 + 2(x + 1)

X 64 + 240X 60 Y 4 + 18040X 56 Y 8 + 452816X 52 Y 12 + 5260060X 48 Y 16 + 1048576X 46 Y 18 + 32253936X 44 Y 20 + 36700160X 42 Y 22 + 112900424X 40 Y 24 + 286261248X 38 Y 26 + 235712080X 36 Y 28 + 749731840X 34 Y 30 + 1374288454X 32 Y 32 + 749731840X 30 Y 34 + 235712080X 28 Y 36 + 286261248X 26 Y 38 + 112900424X 24 Y 40 + 36700160X 22 Y 42 + 32253936X 20 Y 44 + 1048576X 18 Y 46 + 5260060X 16 Y 48 + 452816X 12 Y 52 + 18040X 8 Y 56 + 240X 4 Y 60 + Y 64

5

(x +1)30 +2(x +1), 2(x +1)2 

6

(x + 1)28 + 2

7

(x + 1)24 + 2

8

(x + 1)28 + 2 + 2(x + 1)3 

X 64 + 112X 60 Y 4 + 4984X 56 Y 8 + 111440X 52 Y 12 + 262144X 50 Y 14 + 1410332X 48 Y 16 + 5505024X 46 Y 18 + 13585264X 44 Y 20 + 44040192X 42 Y 22 + 119045192X 40 Y 24 + 234881024X 38 Y 26 + 449757392X 36 Y 28 + 789053440X 34 Y 30 + 979654214X 32 Y 32 + 789053440X 30 Y 34 + 449757392X 28 Y 36 + 234881024X 26 Y 38 + 119045192X 24 Y 40 + 44040192X 22 Y 42 + 13585264X 20 Y 44 + 5505024X 18 Y 46 + 1410332X 16 Y 48 + 262144X 14 Y 50 +111440X 12 Y 52 +4984X 8 Y 56 +112X 4 Y 60 +Y 64

9

(x + 1)28 + 2 + 2(x + 1)2 

X 64 +112X 60 Y 4 +4984X 56 Y 8 +4096X 54 Y 10 +111440X 52 Y 12 + 258048X 50 Y 14 + 1344796X 48 Y 16 + 5419008X 46 Y 18 + 14109552X 44 Y 20 + 44388352X 42 Y 22 + 117210184X 40 Y 24 + 234332160X 38 Y 26 + 453427408X 36 Y 28 + 789340160X 34 Y 30 + 975066694X 32 Y 32 + 789340160X 30 Y 34 + 453427408X 28 Y 36 + 234332160X 26 Y 38 + 117210184X 24 Y 40 + 44388352X 22 Y 42 + 14109552X 20 Y 44 + 5419008X 18 Y 46 + 1344796X 16 Y 48 + 258048X 14 Y 50 + 111440X 12 Y 52 + 4096X 10 Y 54 + 4984X 8 Y 56 + 112X 4 Y 60 + Y 64

10

(x + 1)28 + 2 + 2(x + 1)2 + 2(x + 1)3 

X 64 + 112X 60 Y 4 + 4984X 56 Y 8 + 144208X 52 Y 12 + 2721052X 48 Y 16 + 31312752X 44 Y 20 + 219708488X 40 Y 24 + 939507920X 36 Y 28 + 1908168262X 32 Y 32 + 939507920X 28 Y 36 + 219708488X 24 Y 40 + 31312752X 20 Y 44 + 2721052X 16 Y 48 + 144208X 12 Y 52 + 4984X 8 Y 56 + 112X 4 Y 60 + Y 64

X 64 + 112X 60 Y 4 + 5496X 56 Y 8 + 154448X 52 Y 12 + 2722076X 48 Y 16 + 30888816X 44 Y 20 + 221665864X 40 Y 24 + 935203024X 36 Y 28 + 1913687622X 32 Y 32 + 935203024X 28 Y 36 + 221665864X 24 Y 40 + 30888816X 20 Y 44 + 2722076X 16 Y 48 + 154448X 12 Y 52 + 5496X 8 Y 56 + 112X 4 Y 60 + Y 64

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Table 3 Cyclic self-dual codes over Z9 of length 27. No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Ideals 3 (x − 1)26 , 3(x − 1) (x − 1)25 , 3(x − 1)2  (x − 1)25 + 6 + 3(x − 1) (x − 1)25 + 3 + 2(x − 1) (x − 1)24 , 3(x − 1)3  (x − 1)24 + 3(x − 1) + 3(x − 1)2 , 3(x − 1)3  (x − 1)24 + 6(x − 1) + 6(x − 1)2 , 3(x − 1)3  (x − 1)23 , (x − 1)4  (x − 1)23 + 3(x − 1)2 , 3(x − 1)4  (x − 1)23 + 6(x − 1)2 , 3(x − 1)4  (x − 1)23 + 3 + 3(x − 1) + 6(x − 1)3  (x − 1)23 + 3 + 3(x − 1) + 6(x − 1)2 + 6(x − 1)3  (x − 1)23 + 6 + 6(x − 1) + 6(x − 1)2 + 3(x − 1)3  (x − 1)23 + 3 + 3(x − 1) + 3(x − 1)2 + 6(x − 1)3  (x − 1)23 + 6 + 6(x − 1) + 3(x − 1)2 + 3(x − 1)3  (x − 1)23 + 6 + 6(x − 1) + 3(x − 1)3 

• n = 27 There are exactly 162 principal ideals and 80 non-principal ideals of Z9 [x]/x27 − 1 which are associated with cyclic self-dual codes over Z9 of length 27. Among then, there are 240 extremal cyclic self-dual codes over Z9 of length 27 with respect to Lee weights. The case of n = 27 is given in Table 3, but it is not an entire list; a complete list of codes of lengths n = 3, 9, and 27 can be found in [11].

4.3. Cyclic self-dual codes over GR(32 , 2) of lengths up to 9 The computation result is obtained using Proposition 3.1 as follows. Theorem 4.3. We find all ideals of GR(32 , 2)[x]/xn − 1 associated with cyclic self-dual codes over GR(32 , 2) of lengths n = 3 and 9. • n=3 There are exactly 2 principal ideals of GR(32 , 2)[x]/x3 − 1 which are associated with cyclic self-dual codes over GR(32 , 2) of length 3. • n=9 There are exactly 18 principal ideals and 2 non-principal ideals of GR(32 , 2)[x]/ x9 − 1 which are associated with cyclic self-dual codes over GR(32 , 2) of length 9. Among then, there are 18 extremal cyclic self-dual codes over GR(32 , 2) of length 9 with respect to Lee weights.

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Table 4 Cyclic self-dual codes over GR(32 , 2) of length 9. No. Ideals 1 3 2 (x − 1)8 , 3(x − 1) 3 (x − 1)7 + 6(x − 1), 3(x − 1)2  4 (x − 1)7 + 3(ξ + 2) + 6ξ(x − 1) 5 (x − 1)7 + 6(ξ + 1) + 3ξ(x − 1) 6 (x − 1)7 + 6 7 (x − 1)7 + 3(2ξ + 1) + 3(ξ + 1)(x − 1) 8 (x − 1)7 + 6ξ + 3(ξ + 2)(x − 1) 9 (x − 1)7 + 3ξ + 6(ξ + 1)(x − 1) 10 (x − 1)7 + 3 + 3(x − 1)

No. 11 12 13 14 15 16 17 18 19 20

129

Ideals (x − 1)7 + 3(ξ + 1) + 3(2ξ + 1)(x − 1) (x −1)6 +6 +3(ξ +2)(x −1) +3(ξ +2)(x −1)2  (x − 1)6 + 6 + 6(x − 1) + 6(x − 1)2  (x − 1)6 + 6 + 6ξ(x − 1) + 6ξ(x − 1)2  (x −1)6 +6 +6(ξ +1)(x −1) +6(ξ +1)(x −1)2  (x − 1)6 + 6 + 3ξ(x − 1) + 3ξ(x − 1)2  (x − 1)6 + 6 + 3(x − 1) + 3(x − 1)2  (x − 1)6 + 6 (x−1)6 +6+3(2ξ+1)(x−1)+3(2ξ+1)(x−1)2  (x −1)6 +6 +3(ξ +1)(x −1) +3(ξ +1)(x −1)2 

The case of n = 9 is given in Table 4. A complete list of codes of length n = 3 can be found in [11].

Acknowledgment We would like to thank the referee for valuable comments, which were very helpful for improving the clarity of our paper. References [1] T. Abualrub, R.B. Oehmke, On the generator of Z4 cyclic codes of length 2e , IEEE Trans. Inf. Theory 49 (2003) 2126–2133. [2] M. Bhaintwal, S.K. Wasan, On quasi-cyclic codes over Zq , Appl. Algebra Eng. Commun. Comput. 20 (2009) 459–480. [3] A.R. Calderbank, N.J.A. Sloane, Modular and p-adic cyclic codes, Des. Codes Cryptogr. 6 (1995) 21–35. [4] S.T. Dougherty, C. Fernandez-Cordoba, Codes over Z2k , gray maps and self-dual codes, Adv. Math. Commun. 5 (2011) 571–588. [5] S.T. Dougherty, T.A. Gulliver, J. Wong, Self-dual codes over Z8 and Z9 , Des. Codes Cryptogr. 41 (2006) 235–249. [6] S.T. Dougherty, S. Ling, Cyclic codes over Z4 of even length, Des. Codes Cryptogr. 39 (2006) 127–153. [7] M. Harada, A. Munemasa, On the classification of self-dual Zk -codes, in: Cryptography and Coding, in: Lect. Notes Comput. Sci., vol. 5921, 2009, pp. 78–90. [8] X. Kai, S. Zhu, On the distances of cyclic codes of length 2e over Z4 , Discrete Math. 310 (2010) 12–20. [9] H.M. Kiah, K.H. Leung, S. Ling, Cyclic codes over GR(p2 , m) of length pk , Finite Fields Appl. 14 (2008) 834–846. [10] H.M. Kiah, K.H. Leung, S. Ling, A note on cyclic codes over GR(p2 , m) of length pk , Des. Codes Cryptogr. 63 (2012) 105–112. [11] Y. Lee, http://www.math.ewha.ac.kr/~yoonjinl/Galoisrings.pdf. [12] S.R. Lopez-Permouth, H. Ozadam, F. Ozbudak, S. Szabo, Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes, Finite Fields Appl. 19 (2013) 16–38. [13] V. Pless, J.S. Leon, J. Fields, All Z4 of type II and length 16 are known, J. Comb. Theory 78 (1997) 32–50. [14] V. Pless, P. Sole, Z. Qian, Cyclic self-dual Z4 -codes, Finite Fields Appl. 3 (1997) 48–69. [15] R. Sobhani, M. Esmaeili, A note on cyclic codes over GR(p2 , m) of length pk , Finite Fields Appl. 15 (2009) 387–391.

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