Asymptotically good ZprZps -additive cyclic codes

Finite Fields and Their Applications 63 (2020) 101633

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Asymptotically good Zpr Zps -additive cyclic codes Ting Yao, Shixin Zhu ∗ , Xiaoshan Kai School of Mathematics, Hefei University of Technology, Hefei 230601, Anhui, PR China

a r t i c l e

i n f o

Article history: Received 12 June 2019 Received in revised form 3 December 2019 Accepted 17 December 2019 Available online xxxx Communicated by Qiang Wang MSC: 94B05 94B15

a b s t r a c t We construct a class of Zpr Zps -additive cyclic codes generated by pairs of polynomials, where p is a prime number. Based on probabilistic arguments, we determine the asymptotic rates and relative distances of this class of codes: the asymptotic s−r Gilbert-Varshamov bound at 1+p2 δ is greater than 12 and the relative distance of the code is convergent to δ, while the rate is convergent to 1+p1s−r for 0 < δ < 1+p1s−r and 1 ≤ r < s. As a consequence, we prove that there exist numerous asymptotically good Zpr Zps -additive cyclic codes. © 2020 Elsevier Inc. All rights reserved.

Keywords: Zpr Zps -additive cyclic codes Random codes Cumulative weight enumerator Gilbert-Varshamov bound Asymptotically good

1. Introduction A sequence of codes C1 , C2 , · · · with length ni of Ci going to infinity is said to be asymptotically good, if both the rate of Ci and the relative distance of Ci are positively bounded from below. A class of codes is said to be asymptotically good if there exist * Corresponding author. E-mail addresses: [email protected] (T. Yao), [email protected] (S. Zhu), [email protected] (X. Kai). https://doi.org/10.1016/j.ffa.2020.101633 1071-5797/© 2020 Elsevier Inc. All rights reserved.

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T. Yao et al. / Finite Fields and Their Applications 63 (2020) 101633

asymptotically good sequences of codes within the class. By using a probabilistic method, linear codes are proved to be asymptotically good in [27]. Özbudak and Solé obtained Gilbert-Varshamov type bounds for linear codes over finite chain rings in [19]. Recently, it is worth mentioning that Shi et al. constructed different kinds of codes which are shown to satisfy the modified Gilbert-Varshamov bound in [20,21]. There is the long-standing question whether the class of cyclic codes are asymptotically good. Additive codes (mixed alphabet codes) were first defined by Delsarte in 1973 in terms of association schemes [7]. Under binary Hamming scheme, any abelian binary propelinear codes are isomorphic to Zγ2 × Zδ4 , where γ and δ are positive integers. In general, any β subgroup of Zα 2 × Z4 is called a Z2 Z4 -additive code. Later, Z2 Z4 -additive codes were generalized to Zpr Zps -additive codes in [2]. Zp Zpk -additive codes and their duality were well-described in [24], and Shi et al. constructed 1-perfect additive codes in the mixed Zp Zp2 . . . Zpk alphabet. In 2014, Abualrub et al. first introduced Z2 Z4 -additive cyclic codes, which are defined as Z4 [x]-submodules of Z2 [x]/xα − 1 × Z4 [x]/xβ − 1 [1]. Recently, Broges et al. generalized the results on Z2 Z4 -additive codes to Zpr Zps -additive cyclic codes in [3]. Many examples of (1 + 2u)-constacyclic codes over Z4 [u]/u2 − 1 whose Z4 images are Z4 -cyclic codes with improved parameters were given in [23]. For a finite group G of order m, any left ideal C of the group ring F G is called a  group code of length m over a finite field F , with the codewords of the form g∈G ag g, where ag ∈ F . Any F G-submodule of (FG)n is called a quasi-group code of index n and co-index m. If G is abelian, then the quasi-group codes are also called quasi-abelian codes [8,28], and if G is cyclic the codes are just quasi-cyclic codes. Quasi-abelian codes and quasi-cyclic codes are proved to be asymptotically good in [5,6,13]. Quasi-cyclic codes of fractional index are showed to be asymptotically good in [11] and [16], respectively. In [14], Ling and Solé showed that the long binary self-dual quasi-cyclic codes are asymptotically good. Shi et al. generalized the result to generalized quasi-cyclic codes in [22], and proved that there exist good self-dual generalized quasi-cyclic codes. Furthermore, Shi et al. proved the existence of asymptotically good additive cyclic codes in [25]. Quasi-twisted codes with constacyclic constituent codes was also well described in [26]. In [4], Bazzi and Mitter proved that the random binary quasi-abelian codes of index 2 and random binary dihedral group codes are asymptotically good by using the result on weights of balanced codes. Later, with the similar method, Martínez-Pérez and Willems proved that self-dual doubly even 2-quasi-cyclic transitive codes are asymptotically good [17]. In [10], Fan and Lin studied the first moment and the second moment of the cumulative weight enumerator of the random code. They gave the thresholds of random quasi-abelian codes and proved that the quasi-abelian codes are asymptotically good, which attain the Gilbert-Varshamov bound. In [15], Liu constructed a class of Z2 Z4 -additive cyclic codes generated by pairs of polynomials, and proved that the Z2 Z4 -additive cyclic codes are asymptotically good. The purpose of this paper is to study the asymptotic properties of the class of codes with the co-index m going to infinity. Inspired by the idea of [10] and [15], we construct a

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class of Zpr Zps -additive cyclic codes with cyclic length m. Using a probabilistic method, we prove that there exists a series of asymptotically good Zpr Zps -additive cyclic codes. The paper is organized as follows. Section 2 gives some definitions and results about Zpr Zps -additive cyclic codes. In Section 3, we construct a class of Zpr Zps -additive cyclic codes generated by pairs of polynomials. In Section 4, we give the main result by investigating the asymptotic properties of the rates and relative distances of the class of codes. In Section 5, we make a conclusion. 2. Preliminaries Let Zpr and Zps be the rings of integers modulo pr and ps , respectively, with 1 ≤ r < s s−1 r−1 and p is a prime. For any a ∈ Zps , b ∈ Zpr , express a = i=0 ai pi and b = j=0 bj pj , where ai , bj ∈ Zp , i = 0, 1, . . . , s − 1, j = 0, 1, . . . , r − 1. Consider the surjective ring homomorphism [3]: π : Zps → Zpr , a → a (mod pr ). Obviously, if i ≥ r, then π(pi ) = 0. We define a multiplication ∗ in Zpr as follows: a ∗ b = π(a)b, where a ∈ Zps and b ∈ Zpr . Then Zpr is a Zps -modulo with the external multiplication given by π. Let β a ∈ Zps and u = (u, u ) = (u0 , u1 , . . . , uα−1 , u0 , u1 , . . . , uβ−1 ) ∈ Zα pr × Zps , we define the external multiplication of a and u as a ∗ u = (π(a)u0 , π(a)u1 , . . . , π(a)uα−1 , au0 , au1 , . . . , auβ−1 ).

(1)

β Definition 2.1. Let p be a prime number. For 1 ≤ r < s, a subgroup C of Zα pr × Zps is called a Zpr Zps -additive code. β Obviously, a Zpr Zps -additive code C is a Zps -submodule of Zα pr × Zps with respect to the external multiplication (1). The first α coordinates of codewords of C are elements of Zpr and the remaining β coordinates are elements of Zps . From the theorem of finite kr−1 β abelian groups, the additive code C, as a subgroup of Zα × pr × Zps , is isomorphic to Zp kr−2 l k0 l0 l1 s−1 Zp2 × · · · × Zpr × Zps × Zps−1 × · · · × Zp . We say that such an additive code C is r−1 of type (α, β; k0 , k1 , . . . , kr−1 ; l0 , l1 , . . . , ls−1 ) and |C| = pK , where K = i=0 (r − i)ki + s−1 j=0 (s − j)lj for 1 ≤ r < s. Let u = (u0 , u1 , . . . , uα−1 , u0 , u1 , . . . , uβ−1 ) and v = (v0 , v1 , . . . , vα−1 , v0 , v1 , . . . , β  vβ−1 ) ∈ Zα pr × Zps , then the inner product of u and v is defined as

u, v = p

s−r

α−1  i=0

 ui vi

+

β−1 

uj vj ∈ Zps .

j=0

Let C be a Zpr Zps -additive code. The dual code of C, denoted by C ⊥ , is defined in the standard way by

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β C ⊥ = {u ∈ Zα pr × Zps | u, v = 0 for all v ∈ C}. β Then C ⊥ is also a Zpr Zps -additive code since C ⊥ is a subgroup of Zα pr × Zps . In [29], Yildiz and Odemis defined a distance-preserving Gray map Φk from Zpk to pk−1 Zp , which is an extended Lee weight to Hamming weight. First, we introduce the new extension of Lee weight on Zpk in the following:

⎧ ⎪ ⎪ ⎨ wtL (x) =

x,

pk−1 , ⎪ ⎪ ⎩ pk − x,

if x ≤ pk−1 , if pk−1 ≤ x ≤ pk − pk−1 , if pk − pk−1 < x ≤ pk − 1. k−1

Definition 2.2. ([29]) Define the Gray map Φk from Znpk to Zpp

n

as

Φk (x0 , x1 , . . . , xn−1 ) = (Φk (x0 ), Φk (x1 ), . . . , Φk (xn−1 )), where xi ∈ Zpk for 0 ≤ i ≤ n − 1. If xi ≤ pk−1 , the image of xi is to put 1 s in the first x coordinates and 0 s in the rest of the coordinates. If xi > pk−1 , then the image of xi is q i + Φk (ri ), where q i = (qi qi qi . . . qi qi qi ) and xi = qi pk−1 + ri for 0 ≤ qi ≤ p − 1 and 0 ≤ ri ≤ pk−1 − 1. Note that the Gray map Φk is a distance-preserving map from (Znpk , dL ) to k−1

(Zpp n , dH ), where dH and dL denote the Hamming distance and the extending Lee distance of two vectors x, y in Znpk , respectively. β Definition 2.3. For all x = (x0 , x1 , . . . , xα−1 ) ∈ Zα pr , y = (y0 , y1 , . . . , yβ−1 ) ∈ Zps , and β N N = pr−1 α + ps−1 β, the Gray-like map Φ : Zα pr × Zps −→ Zp is defined by

Φ(x, y) = (Φr (x0 ), Φr (x1 ), . . . , Φr (xα−1 ), Φs (y0 ), Φs (y1 ), . . . , Φs (yβ−1 )).

(2)

Note that the Gray-like map Φ is not bijective in general, because it is not surjective. However, the Gray-like map Φ defined in (2) is a distance-preserving map, β N which transforms the Lee weight in Zα pr × Zps to the Hamming distance in Zp , where r−1 s−1 N = p α + p β. β Let u = (u1 , u2 ) ∈ Zα pr × Zps , the weight of u is defined by wt(u) = wt(u1 , u2 ) = wtL (u1 ) + wtL (u2 ), β where wtL (u1 ) and wtL (u2 ) are the extended Lee weights of u1 ∈ Zα pr and u2 ∈ Zps , respectively. Let C be a Zpr Zps -additive code, the minimum weight of the code C is defined by wt(C) = min{wt(c) | c ∈ C\{0}}. The ratios

R(C) =

d K and Δ(C) = N N

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are called the rate and the relative minimum distance of C, respectively, where d = r−1 s−1 min0=c∈C wt(c), K = i=0 (r − i)ki + j=0 (s − j)lj and N = pr−1 α + ps−1 β. Recall that 1 ≤ r < s, let χ be an additive group homomorphism from Zpr into Zps as follows: χ : Zpr → Zps a → ps−r a. The image of the homomorphism is ps−r Zps = {0, ps−r , 2ps−r , . . . , (pr − 1)ps−r } ⊂ Zps . β Definition 2.4. A non-empty subset C of Zα pr ×Zps is called a Zpr Zps -additive cyclic code if (i) C is an additive code, and (ii) for any codeword c = (a0 , a1 , . . . , aα−1 , b0 , b1 , . . . , bβ−1 ) ∈ C, its cyclic shift ρ(c) = (aα−1 , a0 , . . . , aα−2 , bβ−1 , b0 , . . . , bβ−2 ) is still in C. Denote by Rα,β = Zpr [x]/xα −1 ×Zps [x]/xβ − 1. There is a bijective map between α Zpr × Zβps and Rα,β given by:

(a0 , . . . , aα−1 , b0 , . . . , bβ−1 ) → (a0 + . . . + aα−1 xα−1 , b0 + . . . + bβ−1 xβ−1 ). β We denote the image of the vector c = (a0 , . . . , aα−1 , b0 , . . . , bβ−1 ) ∈ Zα pr × Zps by c(x), that is, c(x) = (a(x), b(x)) ∈ Rα,β , where a(x) = a0 + . . . + aα−1 xα−1 ∈ Zpr [x]/xα − 1 and b(x) = b0 + . . . + bβ−1 xβ−1 ∈ Zps [x]/xβ − 1. Let f (x) ∈ Zps [x] and (g(x), h(x)) ∈ Rα,β . We define

f (x) ∗ (g(x), h(x)) = (π(f (x))g(x), f (x)h(x)) ∈ Rα,β , where π(f (x)) ≡ f (x) (mod pr ), π(f (x))g(x) ∈ Rα = Zpr [x]/xα − 1, and f (x)h(x) ∈ Rβ = Zps [x]/xβ − 1. Therefore, Rα,β is a Zps [x]-module with respect to the multiplication above. 3. A class of Zpr Zps -additive cyclic codes In this section, we construct a new class of Zpr Zps -additive cyclic codes, which is the Zps [x]-submodule of Rα,β . For the rest of the discussion, we always assume that α = β = m and gcd(m, p) = 1. From Section 2, we denote  Rm = Zpr [x]/xm − 1, Rm = Zps [x]/xm − 1,

and ps−r Zps = {0, ps−r , 2ps−r , . . . , (pr − 1)ps−r } ⊂ Zps .

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T. Yao et al. / Finite Fields and Their Applications 63 (2020) 101633

m A non-empty subgroup C of Zm pr × Zps is a Zpr Zps -additive cyclic code if and only if C  is a Zps [x]-submodule of Rm × Rm .  Consider a class of Zps [x]-submodule of Rm × Rm . Let

p

s−r

 Rm

=



a (x) =

m−1 

ai xi

∈

 Rm

i=0

  s−r   p  ai , 

 which is an ideal of Rm . From additive group homomorphism χ defined previously, we have a Zps [x]-module  isomorphism from Rm onto ps−r Rm :

Rm → p

s−r

 Rm ,

m−1  i=0

ai x → i

m−1 

ps−r ai xi , where ai ∈ Zpr .

(3)

i=0

  Hence, Rm and ps−r Rm are Zpr [x]-modules. Therefore, Rm × ps−r Rm is a Zps [x]-sub module of Rm × Rm .  Any element of Rm × ps−r Rm can be uniquely represented as (a(x), ps−r b(x)), where m−1 m−1 i a(x) = i=0 ai x , b(x) = i=0 bi xi ∈ Rm . Therefore, for (a(x), b(x)) ∈ Rm × Rm , let  Ca,b = {(f (x)a(x), ps−r f (x)b(x)) ∈ Rm × ps−r Rm | f (x) ∈ Rm }.

(4)

 Then Ca,b is a Zpr Zps -additive cyclic code in Rm × Rm generated by one element (a(x), b(x)). As Ca,b is an Rm -module, the dimension of Ca,b is denoted by dim Ca,b over a Zp -vector space.

Theorem 3.1. Given any (a(x), b(x)) ∈ Rm × Rm . Let ga,b (x) = gcd(a(x), b(x), xm − 1)

and ha,b (x) =

xm − 1 . ga,b (x)

Then (a(x), b(x)) induces an Rm -homomorphism  τa,b : Rm → Rm × ps−r Rm

f (x) → (f (x)a(x), ps−r f (x)b(x)), and the following hold (i) The image im(τa,b ) = Ca,b , where Ca,b is defined in Eq. (4); (ii) The kernel ker(τa,b ) = ha,b (x)Rm , and dim Ca,b = deg(ha,b (x)); (iii) τa,b induces an isomorphism ga,b (x)Rm ∼ = Ca,b ; in particular,  Ca,b = {(f (x)a(x), ps−r f (x)b(x)) ∈ Rm × ps−r Rm | f (x) ∈ ga,b (x)Rm }.

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Proof. (i) Obviously, the image im(τa,b ) = Ca,b for τa,b is an Rm -homomorphism. (ii) For f (x) ∈ Rm , f (x) ∈ ker(τa,b ) if and only if

f (x)a(x) ≡ 0 (mod xm − 1), f (x)b(x) ≡ 0 (mod xm − 1),

which f (x) gcd(a(x), b(x)) ≡ 0 (mod xm − 1) in Zpr [x]. Hence, f (x) ≡

implies that  m x −1 xm −1 0 mod gcd(a(x),b(x),x m −1) . It follows that f (x) ∈ ha,b (x)Rm , where ha,b (x) = g a,b (x) and dim Ca,b = dim Rm − dim ker(τa,b ) = m − deg(ga,b (x)) = deg(ha,b (x)). (iii) Since gcd(m, p) = 1, we have Rm = ga,b (x)Rm ⊕ ha,b (x)Rm . Then the above Rm -homomorphism τa,b induces an Rm -module isomorphism: τ a,b : ga,b (x)Rm → Ca,b c(x) → (c(x)a(x), ps−r c(x)b(x)). So, (iii) is proved. 2 To obtain good asymptotic properties, we construct a suitable class of Zpr Zps -additive cyclic codes on which the probabilistic methods can work well. Instead of considering the whole ring Rm = Zpr [x]/xm − 1, we consider the augmentation ideal Jm of Rm ;   and take the corresponding ideal ps−r Jm of Rm = Zps [x]/xm − 1. In the following, we construct a class of Zpr Zps -additive cyclic codes Ca,b generated by the polynomial pair  (a(x), ps−r b(x)) ∈ Jm × ps−r Jm . Recall that p is a prime, and gcd(m, p) = 1, we have that xm − 1 = (x − 1)(xm−1 + . . . + x + 1) and Rm = Zpr [x]/xm − 1 = xm−1 + . . . + x + 1Rm ⊕ x − 1Rm . In the following, we consider the ideal of Rm generated by x − 1 as follows: Jm = x − 1Rm , which is called the augmentation ideal of Rm in the representation theory of finite groups.    Similarly, let Jm = x − 1Rm be the ideal of Rm = Zps [x]/xm − 1. Moreover,    , ps−r Jm = {ps−r f (x) | f (x) ∈ Jm } = ps−r (x − 1)Rm

and there is an Rm -module isomorphism from Eq. (3):  Jm × Jm → Jm × ps−r Jm

(a(x), b(x)) → (a(x), ps−r b(x)).

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In the light of Theorem 3.1, for a(x), b(x) ∈ Jm ⊆ Rm , the Zpr Zps -additive cyclic codes Ca,b can be written as   Ca,b = {(c(x)a(x), ps−r c(x)b(x)) ∈ Rm × ps−r Rm | c(x) ∈ Jm } ⊆ Jm × ps−r Jm . −1 −1 Let xx−1 = f1 (x)f2 (x) · · · ft (x) be an basic irreducible decomposition of xx−1 in Zpr [x], where f1 (x), f2 (x), . . . , ft (x) are distinct basic irreducible polynomials in Zpr [x]. m

m

Lemma 3.2. Let notations be as above, (a(x), b(x)) ∈ Jm × Jm . Then dim Ca,b ≤ m − 1. Moreover, dim Ca,b < m − 1 if and only if there is an irreducible factor fi (x)(1 ≤ i ≤ t) m −1 in Zpr [x] such that fi (x) | a(x) and fi (x) | b(x). of xx−1 Proof. Since Jm = x − 1Rm , we have that (x − 1) | ga,b (x), where ga,b (x) = gcd(a(x), b(x), xm − 1). Therefore, ga,b (x)Rm ⊆ x − 1Rm = Jm . According to Theorem 3.1, dim Ca,b = dimga,b (x)Rm ≤ dim Jm = m − 1. Hence, dim Ca,b = m − 1 if and only if ga,b (x) = gcd(a(x), b(x), xm − 1) = x − 1. It follows m −1 that there is no irreducible factor fi (x)(1 ≤ i ≤ t) of xx−1 such that fi (x) | a(x) and fi (x) | b(x). The result follows. 2 Lemma 3.3. Let lm = min{deg(f1 (x)), . . . , deg(ft (x))} in Zpr [x], and km be an integer with lm ≤ km ≤ m − 1. Then any non-zero ideal of Rm contained in Jm is of dimension at least lm , and the number of the ideals with dimension km contained in Jm is at most km m lm . Proof. Every irreducible ideal contained in Jm corresponds exactly to one irreducible m −1 divisor fi (x)(1 ≤ i ≤ t) of xx−1 , which implies that the dimension of the ideal is equal to deg(fi (x)(1 ≤ i ≤ t). Therefore, the minimal dimension of the ideal contained in Jm m is lm . Any km -dimension ideal contained in Jm is a sum of at most klm irreducible ideals. Therefore, the number of the km -dimension ideals contained in Jm is at most the partial m    klm Nm sum of i=1 i , where Nm is the number of the irreducible ideals contained in Jm m m and  klm  denotes the largest integer which is not larger than klm . Since Nm < m, by km km  lm  Nm  scaling, we have that the partial sum i=1 is less than m lm . 2 i 4. Random Zpr Zps -additive cyclic codes In this section, keeping the notations in Section 3, we view the sets Jm × Jm and   Jm × ps−r Jm as probability spaces of Rm × Rm and Rm × ps−r Rm , respectively, whose samples are obtained with equal probability. By probabilistic method, we determine the asymptotic rates and relative distance of the class of Zpr Zps -additive cyclic codes. Recall that, for any prime power q, the function αq (x) = 1 − hq (x) = 1 − x logq (q − 1) + x logq x + (1 − x) logq (1 − x)

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is called the q-ary asymptotic Gilbert-Varshamov bound, or GV-bound in short for 0 < x ≤ 1 − q −1 , q ≥ 2, where hq (x) is called the q-ary entropy with 0 logq 0 = 0 conventionally. Then αq (x) is a strictly decreasing convex function in the interval [0, 1 − q −1 ] with αq (0) = 1 and αq (1 − q −1 ) = 0, please refer to [12, Sec. 2.10.6]. For convenience, we always denote that q = pr . In the following, we always assume that δ is a positive 1 −1 real number such that 0 < δ < 1+p2s−r · h−1 q (1/2) < 1+ps−r for hq (1/2) < 1/2, and s−r

s−r

αq ( 1+p2 δ) = 1 − hq ( 1+p2 δ) > 12 with 1 ≤ r < s.  For each sample (a(x), ps−r b(x)) ∈ Jm × ps−r Jm with (a(x), b(x)) ∈ Jm × Jm , we have a Zpr Zps -additive cyclic code with cyclic length m as follows:  Ca,b = {(c(x)a(x), ps−r c(x)b(x)) ∈ Rm × ps−r Rm | c(x) ∈ Jm },

(5)

 which is viewed as a random code over the probability space Jm × ps−r Jm . Hence, the relative distances Δ(Ca,b ) of Ca,b and the dimension dim(Ca,b ) of Ca,b are random variables over the probability space. We are concerned with the asymptotic behavior of the probabilities Pr(dim Ca,b = m − 1) and Pr(Δ(Ca,b ) > δ). Let |S| be the cardinality of any set S. Let NCa,b (δ) be the number of non-zero codewords of Ca,b with relative weight at most δ, i.e.

     s−r r−1 s−r   NCa,b (δ) = {c(x) ∈ Jm 1 ≤ wt(c(x)a(x), p c(x)b(x)) ≤ p (1 + p )mδ}. Then, NCa,b (δ) is a non-negative integral random variable defined over the probability  space Jm × ps−r Jm . According to [9], we call NCa,b (δ) the cumulative weight enumerator of the random code Ca,b . From Theorem 4.2 in [10], to obtain the asymptotic properties of Ca,b , we need to calculate the expectation of NCa,b (δ). We write NCa,b (δ) as a sum of Bernoulli random  variable over the probability space Jm × ps−r Jm . Recall that a random variable is called a Bernoulli variable if it takes the value 1 with probability p and the value 0 with probability 1 − p. For any c(x) ∈ Jm , and samples  (a(x), ps−r b(x)) ∈ Jm × ps−r Jm , we have the following indicator random variable:

Xc =

1, 0,

1 ≤ wt(c(x)a(x), ps−r c(x)b(x)) ≤ pr−1 (1 + ps−r )mδ; otherwise.

Let Ic = {c(x)a(x) ∈ Rm | a(x) ∈ Jm } be the ideal of Rm generated by c(x) for c(x) ∈ Jm , and denote its dimension by dc , that is Ic = c(x)Rm ⊆ Jm and dim Ic = dc .

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T. Yao et al. / Finite Fields and Their Applications 63 (2020) 101633

Lemma 4.1. Let the notations be defined as above. For Ic × Ic ⊆ Rm × Rm , let (Ic ×   r−1 s−r Ic )≤p (1+p )mδ = (p1 (x), p2 (x)) ∈ Ic × Ic | wt(p1 (x), p2 (x)) ≤ pr−1 (1 + ps−r )mδ . Then s−r   (Ic × Ic )≤pr−1 (1+ps−r )mδ  ≤ q 2dc hq ( 1+p2 δ) .

Proof. Note that the length of the code Ic × Ic is 2m, and dim(Rm × Rm ) = 2m. Then the dimension of Ic × Ic is 2dc . The fraction of (1 + ps−r )mδ over the length 2m is s−r s−r (1+ps−r )mδ = (1+p2 )δ , with 0 < δ < 1+p1s−r and hq ( (1+p2 )δ ) < 12 . According to [10, 2m   1+ps−r r−1 s−r Corollary 3.5], we have that (Ic × Ic )≤p (1+p )mδ  ≤ q 2dc hq ( 2 δ) . 2 Lemma 4.2. With the above notations, then E(Xc ) ≤ q −2dc αq (

1+ps−r 2

δ)

.

  Proof. Since ps−r Jm ⊂ Rm , we have the following ideal    Ic = ps−r c(x)Rm = {ps−r c(x)b(x) ∈ Rm | b(x) ∈ Jm } ⊆ ps−r Jm .  By the Rm -module isomorphism from Jm × Jm to Jm × ps−r Jm , then Ic ∼ = Ic and  dim Ic = dim Ic = dc .  , let For Ic × Ic ⊆ Rm × Rm

(Ic × Ic )≤p

r−1

(1+ps−r )mδ

= {(p1 (x), ps−r p2 (x)) ∈ Ic × Ic | wt(p1 (x), ps−r p2 (x)) ≤ pr−1 (1 + ps−r )mδ)}.

Then, we have |(Ic × Ic )≤p (1+p E(Xc ) = Pr(Xc = 1) = |Ic × Ic | r−1

s−r

)mδ

|−1

,

for Xc is a 0-1-variable, where E(Xc ) means the expectation of Xc . For p1 (x), p2 (x) ∈ Rm , from the extended Lee weight on Zpk , we have that wtL (ps−r p2 (x)) = ps−r wtL (p2 (x)),  where wtL (ps−r p2 (x)) and wtL (p2 (x)) are computed in Rm and Rm , respectively. Therefore,

wt(p1 (x), ps−r p2 (x)) ≥ wt(p1 (x), p2 (x)). By Lemma 4.1 and |(Ic × Ic )≤p

r−1

|(Ic × Ic )≤p (1+p E(Xc ) ≤ |Ic × Ic | r−1

(1+ps−r )mδ

s−r

)mδ

|

| ≤ |(Ic × Ic )≤p 1+ps−r

q 2dc hq ( 2 ≤ q 2dc

δ)

r−1

(1+ps−r )mδ

= q −2dc αq (

|, we obtain

1+ps−r 2

δ)

.

2

T. Yao et al. / Finite Fields and Their Applications 63 (2020) 101633

Lemma 4.3. Let 0 < δ < Lemma 3.3. Then

1 1+ps−r

11

and Ca,b be as in Eq. (5). Let lm be as defined in

Pr(Δ(Ca,b ) ≤ δ) ≤

m−1 

1+ps−r 2

q −2j( 2 −hq ( 1

δ)−

logq m 2lm )

.

j=lm

  Proof. Let X = c(x)∈Jm Xc , then X0 = 0 and NCa,b (δ) = c(x)∈Jm Xc = X. Note that X stands for the number of c(x) ∈ Jm such that the non-zero codeword (c(x)a(x), ps−r c(x)b(x)) of Ca,b with relative weight at most pr−1 (1 + ps−r )mδ. Then Pr(Δ(Ca,b ) ≤ δ) = Pr(X > 0). By the Markov’s inequality in [18, Theorem 3.1], we have Pr(X > 0) ≤ E(X). In the  following, we are going to estimate the expectation E(X) = c(x)∈Jm E(Xc ).    For any ideal I of Jm , we denote I ∗ = c(x) ∈ I  Ic = I and dim I = dim Ic = dc . Then 

Jm =

I ∗,

0=I≤Jm

where the subscript “0 = I ≤ Jm ” means that I runs over the non-zero ideals contained in Jm . Therefore, lm ≤ dim I ≤ m − 1 from Lemma 3.3, and E(X) = E







 Xc =



E(Xc ) =

0=I≤Jm c(x)∈I ∗

c(x) ∈Jm

m−1 





E(Xc ).

j=lm 0 = I ≤ Jm c(x)∈I ∗ dim I = j

For I ≤ Jm with dim I = j, |I ∗ | ≤ |I| = q j . By Lemma 4.2, we have 

E(Xc ) ≤

c(x)∈I ∗



q −2jαq (

1+ps−r 2

δ)

c(x)∈I ∗

≤ q j · q −2jαq ( = q −j+2jhq (

1+ps−r 2

1+ps−r 2

δ)

δ)

.

   j In the light of Lemma 3.3, we have {I ≤ Jm  dim I = j} ≤ m lm and logq m ≤ j logq m . lm

Therefore, E(X) ≤

m−1 

j

m lm · q −j+2jhq (

1+ps−r 2

δ)

j=lm

=

m−1  j=lm

q −j+2jhq (

1+ps−r 2

j δ)+ lm logq m

T. Yao et al. / Finite Fields and Their Applications 63 (2020) 101633

12

=

m−1 

q −2j( 2 −hq ( 1

1+ps−r 2

δ)−

logq m 2lm )

2

.

j=lm

According to [4, Lemma 2.6], there exist positive integers m1 , m2 , . . . satisfying that gcd(mi , p) = 1, mi → ∞ and lim

i→∞

logq mi = 0, lmi

where lmi is the minimal degree of the irreducible divisors of Lemma 3.3. For each mi , let

(6)

xmi −1 x−1

as defined in

 Ca,b = {(c(x)a(x), ps−r c(x)b(x)) ∈ Rmi × ps−r Rm | c(x) ∈ Jmi } i (i)

(7)

be the random Zpr Zps -additive cyclic code with cyclic length mi over the probability  space Jmi × ps−r Jm as in Eq. (5). i s−r

Theorem 4.4. Let notations be as above. If αq ( 1+p2

δ) >

1 2

for 0 < δ <

1 1+ps−r ,

then

(i)

lim Pr(Δ(Ca,b ) > δ) = 1.

i→∞ s−r

s−r

Proof. If αq ( 1+p2 δ) > 12 , then 12 − hq ( 1+p2 δ) > 0. When mi → ∞, there exists a positive real number λ and an integer n such that λ≤

logq mi 1 1 + ps−r − hq ( δ) − , ∀ i > n. 2 2 2lmi

By Lemma 4.3, (i)

lim Pr(Δ(Ca,b ) ≤ δ) ≤ lim

i→∞

i→∞

m i −1  j=lmi

q −2jλ ≤ lim

i→∞

m i −1  j=lmi

≤ lim mi q −2lmi λ = lim q i→∞

Since lim

i→∞

logq mi 2lmi

q −2lmi λ

−2lmi (λ−

logq mi 2lm i

)

i→∞

. (i)

= 0, it follows that lim lmi = ∞, so we have that lim Pr(Δ(Ca,b ) ≤ i→∞

(i)

i→∞

(i)

δ) = 0. Then lim Pr(Δ(Ca,b ) > δ) = 1 − lim Pr(Δ(Ca,b ) ≤ δ) = 1. 2 i→∞

i→∞

(i)

Theorem 4.5. Let m1 , m2 , . . . be positive integers satisfying Eq. (6), and Ca,b be as defined in Eq. (7). Then (i)

lim Pr(dim Ca,b = mi − 1) = 1.

i→∞

T. Yao et al. / Finite Fields and Their Applications 63 (2020) 101633

Proof. Let Then

xmi −1 x−1

13

= f1 (x)f2 (x) . . . fti (x) be an basic irreducible decomposition in Zpr [x].

Rmi = Zpr [x]/xmi − 1 ∼ = Zpr [x]/x − 1 × Zpr [x]/f1 (x) × · · · × Zpr [x]/fti (x). Furthermore, Jmi = x − 1Rmi ∼ = Zpr [x]/f1 (x) × · · · × Zpr [x]/fti (x), which implies that there exists a surjective homomorphism: (j) θm : Jmi → Zpr [x]/fj (x) i

c(x) → c(x) (mod fj (x)) for j = 1, 2, . . . , ti . By the Chinese Remainder Theorem, we have the following isomorphism: θm : Jm → Zpr [x]/f1 (x) × · · · × Zpr [x]/ft (x) (1) (t) c(x) → (θm (c(x)), . . . , θm (c(x))). (i)

(j)

(j)

By Lemma 3.2, dim Ca,b = m − 1 if and only if (θmi (a(x)), θmi (b(x))) = (0, 0) in Zpr [x]/fj (x) × Zpr [x]/fj (x) for 1 ≤ j ≤ ti . Let deg fj (x) = dj and q = pr , then (j) (j) |Zpr [x]/fj (x)| = |Fqdj |. Therefore, the probability of (θmi (a(x)), θmi (b(x))) = (0, 0) is q 2dj −1 q 2dj

= 1 − q −2dj , which is randomly independent of the choice of a(x) and b(x) for 1 ≤ j ≤ ti . Therefore, (i)

Pr(dim Ca,b = mi − 1) =

ti 

(1 − q −2dj ).

j=1

For any dj ≥ lmi , j = 1, 2, . . . , ti , we have ti ≤

mi −1 lmi

≤

mi lmi

. Thus,

mi

Pr(dim Ca,b = mi − 1) ≥ (1 − q −2lmi ) lmi (i)

q 2lmi ·

= (1 − q −2lmi ) Since lim

i→∞

logq mi lmi

mi 2lm i

lm q i

= 0, which means that lim lmi = ∞. Then i→∞

log lm logq mi mi −lmi (2− lm + lqm i ) i i = lim q = 0. 2l m i→∞ lm q i→∞ i i

lim

.

T. Yao et al. / Finite Fields and Their Applications 63 (2020) 101633

14

Note that (1 − q −2lmi )q

2lmi

> 14 , we have m

(i) lim Pr(dim Ca,b i→∞

i 2lm 1 = mi − 1) ≥ lim ( ) lmi q i = 1. i→∞ 4

2

According to Theorems 4.4 and 4.5, we have the following corollary. s−r

Corollary 4.6. Let δ be a positive real number such that αq ( 1+p2 δ) > 12 for 0 < δ < 1 r s 1+ps−r . Then there is a sequence of Zp Zp -additive cyclic codes C1 , C2 , . . . with cyclic length mi → ∞ and the following hold: (i) lim R(Ci ) = 1+p1s−r , 1 ≤ r < s; i→∞

(ii) Δ(Ci ) > δ for all i = 1, 2, . . .. dim Ci mi −1 Proof. (i) By the definition of the rate of Ci , R(Ci ) = (1+p = (1+p from s−r )m s−r )m i i mi −1 1 Theorem 4.5. Then lim R(Ci ) = lim (1+ps−r )mi = 1+ps−r for 1 ≤ r < s. i→∞

i→∞

(ii) From Theorem 4.4, there exists a Zpr Zps -additive cyclic code Ci with cyclic length mi such that Δ(Ci ) > δ for i > n. Delete the first n codes, and renumber the remaining codes, then we obtain the results. 2 5. Conclusion In this paper, we constructed a class of Zpr Zps -additive cyclic codes, which is generated by pairs of polynomials with cyclic length m. By using a probabilistic method, we determined the asymptotic rates and relative distances of this class of codes. Consequently, we proved that there exist numerous asymptotically good Zpr Zps -additive cyclic s−r codes with αq ( 1+p2 δ) > 12 for 0 < δ < 1+p1s−r , p ≥ 2 and 1 ≤ r < s. The study of the asymptotic properties of cyclic codes is an interesting problem in coding theory. We believe that the asymptotic properties of other families of codes are worth studying in the future. Acknowledgments The authors thank the editor and anonymous referees for their valuable suggestions and comments which have highly improved this paper. This research is supported by the National Natural Science Foundation of China under Grant Nos. 61772168 and 61972126. References [1] T. Abualrub, I. Siap, H. Aydin, Z2 Z4 -additive cyclic codes, IEEE Trans. Inf. Theory 60 (3) (2014) 1508–1514. [2] I. Aydogdu, I. Siap, On Zpr Zps -additive codes, Linear Multilinear Algebra 63 (10) (2015) 2089–2102. [3] J. Borges, C. Fernández-Córdoba, R. Ten-Valls, On Zpr Zps -additive cyclic codes, Adv. Math. Commun. 12 (1) (2018) 169–179.

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