Several Classes of p-ary Linear Codes with Few Weights

arXiv:1612.00915v1 [cs.IT] 3 Dec 2016

Several Classes of p-ary Linear Codes with Few Weights ∗ Rongsheng Wu School of Mathematical Sciences, Anhui University, Hefei, 230601, P. R. China Minjia Shi† Key Laboratory of Intelligent Computing & Signal Processing, Ministry of Education, Anhui University, No. 3 Feixi Road, Hefei Anhui Province 230039, P. R. China, National Mobile Communications Research Laboratory, Southeast University, 210096, Nanjing, P. R. China and School of Mathematical Sciences of Anhui University, Anhui, 230601, P. R. China Patrick Sol´e CNRS/LAGA, University of Paris 8, 2 rue de la libert´e, 93 Saint-Denis, France

Abstract: Few weights codes have been an important topic of coding theory for decades. In this paper, a class of two-weight and three-weight codes for the homogeneous metric over the chain ring R = Fp + uFp + · · · + uk−1 Fp are constructed. These codes are defined as trace codes. They are shown to be abelian. Their homogeneous weight distributions are computed by using exponential sums. In particular, we give a necessary and sufficient condition of the optimality for their Gray images by using the Griesmer bound in the two-weight case, and the information about the dual homogeneous distance is also given. In addition, the codewords of these codes have This research is supported by National Natural Science Foundation of China (61672036), Technology Foundation for Selected Overseas Chinese Scholar, Ministry of Personnel of China (05015133) and the Open Research Fund of National Mobile Communications Research Laboratory, Southeast University (2015D11) and Key projects of support program for outstanding young talents in Colleges and Universities (gxyqZD2016008). ∗

1

been turn out to be minimal, it is significant to obtain secret sharing schemes with interesting access structures. Keywords: Two-weight codes; Three-weight codes; homogeneous distance; optimality

1

Introduction

Let q be a power of prime number p. A q-ary code of length n is a subset of Fnq , and if it is a vector space, it is said to be linear. Let R = Fp + uFp + · · · + uk−1 Fp be a finite local ring, where p is odd prime number and k ≥ 2 a positive integer. Then a code of length n over R is a subset of Rn and it is said to be linear if it is a module. A nonzero code in R is called a two-weight code (resp. three-weight code) if all of its nonzero codewords have two weights (resp. three weights). Two-weight codes and three-weight codes form a class of combinatorial codes which are closely related to combinatorial designs, finite geometry, and graph theory. Information on them can be found in [4, 5]. Some interesting two-weight and threeweight codes were presented in [6, 9, 11]. Recently, the authors in [15, 16, 17] have constructed several infinite family of binary and p-ary few weights codes from trace codes over F2 + uF2, Fp + uFp , and a non-chain ring, respectively. In the present paper, following this trend, we use trace function over a larger ring R defined above. Although most of previous works on two-weight and three-weight codes were done on cyclic codes and cyclotomy [2], the codes we construct here are provably abelian but perhaps not cyclic. Their coordinate places are indexed by the group of units of an algebraic extension of a finite ring. Their homogeneous weight distributions are determined by using character sums. Through the use of a Gray map, we obtain an infinite family of p-ary abelian two-weight and three-weight codes. As for the two-weight case, they are shown to be optimal for given length and dimension by the application of the Griesmer bound under some condition [10]. The manuscript is organized as follows. Section 2 fixes some notations and definitions for this paper. Section 3 presents the main results. The optimality and the dual homogeneous distance are discussed in Section 4. Section 5 determines the support structure of their Gray images, and the application to secret sharing schemes is given. Section 6 summarizes this paper.

2

2 2.1

Preliminaries The ring extension of R

Let R = Fpm + uFpm + · · · + uk−1Fpm , which is a ring extension of R of degree m, and m is a positive integer. There is a generalized Trace function denoted by T r, and defined as T r(a0 + a1 u + · · · + ak−1 uk−1) = tr(a0 ) + tr(a1 )u + · · · + tr(ak−1 )uk−1, for all ai ∈ Fpm , i = 0, 1, . . . , k − 1. Here tr() denotes the standard trace of Fpm down to Fp . Let M denote the maximal ideal of R, we can write M as M = (u) = {a1 u + a2 u2 + · · · + ak−1 uk−1 : ai ∈ Fpm , i = 1, 2, . . . , k − 1}. Let R∗ denote the group of units in R, i.e., R∗ = {a0 + a1 u + · · · + ak−1 uk−1 : a0 ∈ F∗pm , ai ∈ Fpm , i = 1, 2, . . . , k − 1}, and it is immediate to check that the order of R∗ is p(k−1)m (pm − 1). We know R∗ is not a cyclic group and R = R∗ ∪ M. Denoting by Q, and N , respectively, the set of squares and nonsquares in Fpm , we define L = Q × Fpm × · · · × Fpm , such that (k−1)m (pm −1) . Thus L is a subgroup of R∗ of index 2. |L| = p 2

2.2

Gray map

Any integer z can be written uniquely as z = p0 (z) + pp1 (z) + p2 p2 (z) + · · · , where k−1 0 ≤ pi (z) ≤ p − 1, i = 0, 1, 2, . . .. The Gray map Φ : R → Fpp is defined as follows: Φ(a) = (b0 , b1 , b2 , . . . , bpk−1 −1 ), where a = a0 + a1 u + · · · + ak−1 uk−1 . Then for all 0 ≤ i ≤ pk−2 − 1, 0 ≤ ǫ ≤ p − 1, we have  k−2  ak−1 + P pl−1 (i)al + ǫa0 , if k ≥ 3, bip+ǫ = l=1  a1 + ǫa0 ,

if k = 2.

For instance, when k = 3, p = 3, we write Φ(a0 + a1 u + a2 u2 ) = (b0 , b1 , b2 , . . . , b8 ). k−2 P pl−1 (i)al = According to the above definition, so 0 ≤ i ≤ 2, 0 ≤ ǫ ≤ 2 and l=1

p0 (i)a1 = ia1 . Then we get

b0 = a2 , b1 = a2 + a0 , b2 = a2 + 2a0 , b3 = a2 + a1 , b4 = a2 + a1 + a0 , b5 = a2 + a1 + 2a0 , b6 = a2 + 2a1 , b7 = a2 + 2a1 + a0 , b8 = a2 + 2a1 + 2a0 . It is easy to extend the Gray map from Rn to Fpp Φ is injective and linear. 3

k−1 n

, and we also know from [18] that

2.3

Homogeneous metric

For x = (x1 , x2 , . . . , xn ), y = (y1 , y2 , . . . , yn ) ∈ Fnp , dH (x, y) = |{i : xi 6= yi }| is called the Hamming distance between x and y and wH (x) = dH (x, 0), the Hamming weight of x. The Hamming weight of a codeword c = (c1 , c2 , . . . , cn ) of Fnp can also be P equivalently defined as wH (c) = ni=1 wH (ci ), where wH (ci ) equals to 0 if and only if ci is a zero element. The homogeneous weight of an element     0, whom (x) = pk−1 ,    (p − 1)pk−2 ,

x ∈ R is define as follows: if x = 0, if x ∈ (uk−1 )\{0}, if x ∈ R\(uk−1 ).

The homogeneous weight of a codeword c = (c1 , c2 , . . . , cn ) of Rn is defined as P whom (c) = ni=1 whom (ci ). For any x, y ∈ R, the homogeneous distance dhom is given by dhom (x, y) = whom (x − y). As was observed in [18], Φ is a distance preserving k−1 isometry from (Rn , dhom ) to (Fpp n , dH ), where dhom and dH denote the homogeneous k−1 and Hamming distance in Rn and Fpp n , respectively. This means if C is a linear code over R with parameters (n, pt , d), then Φ(C) is a linear code of parameters [pk−1 n, t, d] over Fp .

2.4

Character sums

Now we present some basic facts about Gauss sums. Let q be a power of a prime p, the canonical additive character ψ of Fq is defined as ψ(c) = e2πitr(c)/p , where c ∈ Fq and tr is the absolute trace function from Fq to Fp . Let χ denote an arbitrary multiplicative character of Fq . Assume q is odd and let η be a quadratic multiplicative character of Fq , which is defined by η(x) = 1, if x is the square of an element of F∗q and η(x) = −1 otherwise. Then Gauss sum is defined by G(χ) =

X

ψ(x)χ(x).

x∈F∗q

We define the following character sums X X Q= ψ(x), N = ψ(x). x∈Q

x∈N

4

By orthogonality of characters [12, Lemma 9], it is easy to check that Q + N = −1. , we get then Noting that the characteristic function of Q is 1+η 2 Q=

G(η) − 1 −G(η) − 1 , N= . 2 2

Let ( ap ) denote the Legendre symbol for a prime p and an integer a. The quadratic Gauss sums are well known [8], that if q = pm , and given as follows: p G(η) = (−1)m−1 (p∗ )m , )p = (−1) where p∗ = ( −1 p

3

p−1 2

p.

The main results

In order to obtain our main results, we introduce the following notations unless otherwise stated: • Ev(a) = (T r(ax))x∈L , where a is an element of the ring R, and Ev() is a evaluation map. • Cu,m,p = {Ev(a)|a ∈ R}, a code of length |L| over R. • N = pk−1|L| = (pm − 1)p(k−1)(m+1) /2. • ℜ(∆) is the real part of ∆. Given a finite abelian group G, a code over R is said to be abelian if it is an ideal of the group ring R[G]. In other words, the coordinates of C are indexed by elements of G and G acts regularly on this set. In the special case when G is cyclic, the code is a cyclic code in the usual sense [12]. Proposition 1. The subgroup of units L acts regularly on the coordinates of Cu,m,p . Proof. For any v ′ , u′ ∈ L the change of variables x 7→ (u′ /v ′ )x permutes the coordinates of Cu,m,p , and maps v ′ to u′. Such a permutation is unique, given v ′ , u′ . The code Cu,m,p is thus an abelian code with respect to the group L. In other words, it is an ideal of the group ring R[L]. As observed in Subsection 2.1, we know L is not a cyclic group, thus Cu,m,p may be not cyclic. 5

PN N yj ), and y = (y , y , . . . , y ) ∈ F , then we define Θ(y) = Let ω = exp( 2πi 1 2 N p j=1 ω . p For convenience, we write θ(a) = Θ(Φ(Ev(a))). It can be verified that θ(sa) = Θ(Φ(Ev(sa))), for any s ∈ F∗p . Before starting the computation of the homogeneous weight enumerator, we first state some auxiliary lemmas. Lemma 1. [17, Lemma1 ] For all y = (y1 , y2 , . . . , yn ) ∈ Fnp , we have p−1 X

Θ(sy) = (p − 1)n − pwH (y).

s=1

According to the above lemma, we can check that for any codeword Ev(a) of Cu,m,p , we have P (p − 1)N − p−1 s=1 θ(sa) . whom (Ev(a)) = p The following lemma is the key to study the Gray image Φ(Cu,m,p ), it guarantees the dimension of Φ(Cu,m,p ) is km. The trace function is nondegenerate here, and the proof is easy, we omit it. Lemma 2. For a fixed element x ∈ L, if for any a, b ∈ R, T r(ax) = T r(bx), then we have a = b. Next, we introduce another lemma, it is important to simplify the operation of the proof of case (b) in Theorem 1. Lemma 3. Let a = a1 u + a2 u2 + · · · + ak−1 uk−1 ∈ M\{0}, where M is the principal ideal generated by the element u of R, x = x0 + x1 u + · · · + xk−1 uk−1 ∈ L, and P Pk−2 ai xk−1−i . Then x1 ,...,xk−2 ∈Fpm ω tr(B) 6= 0 if and only if ai = 0 for i = B = i=1 P 1, 2, . . . , k − 2. Furthermore, we have x1 ,...,xk−2 ∈Fpm ω tr(B) = p(k−1)m when a1 = a2 = · · · = ak−2 = 0. Proof. We assume that there exists a aj 6= 0, for j ∈ {1, 2, . . . , k − 2}, then we just P need to consider the term ω tr(aj xk−1−j ) , this term is obviously equals to zero, xk−1−j ∈Fpm P so ω tr(B) = 0, a contradiction. If ai = 0 for i = 1, 2, . . . , k − 2, it is easy x1 ,...,xk−2 ∈Fpm P to check that ω tr(B) = p(k−1)m . The proof is completed. x1 ,...,xk−2 ∈Fpm

Now, we discuss the homogeneous weight of the codewords in Cu,m,p from two aspects. If m is even and p is odd prime, we get three-weight codes. On the other hand, we get two-weight codes when m is odd and p ≡ 3 (mod 4). 6

3.1

m is even

Theorem 1. For a ∈ R. Assume m is even and p ≡ 1 (mod 4). (a) If a = 0, then whom (Ev(a)) = 0; (b) If a ∈ M\{0}, then N, if a ∈ M\{ak−1 uk−1 : ak−1 ∈ Fpm }, then whom (Ev(a)) = p−1 p if a = a′k−1 uk−1, where a′k−1 ∈ F∗pm , then if m (N + (p(k−1)(m+1) (p 2 + 1))/2), a′k−1 ∈ Q, then whom (Ev(a)) = p−1 p m a′k−1 ∈ N , then whom (Ev(a)) = p−1 (N − (p(k−1)(m+1) (p 2 − 1))/2); p (c) If a ∈ R∗ , then whom (Ev(a)) =

p−1 N. p

Proof. Since m is even, it is easy to verify that s ∈ F∗p is always a square in Fpm . Thus θ(sa) = θ(a), for any s ∈ F∗p . Let x = x0 + x1 u + · · · + xk−1 uk−1 ∈ L, where x0 ∈ Q, xi ∈ Fpm for i = 1, 2, . . . , k − 1 throughout the process of proof in this theorem. (a) If a = 0, then Ev(a) = (0, 0, · · · , 0). So whom (Ev(a)) = 0. | {z } |L|

2

(b) Let a = a1 u + a2 u + · · · + ak−1 uk−1 ∈ M\{0}, by a direct calculation we get T r(ax) = tr(a1 x0 )u + tr(a1 x1 + a2 x0 )u2 + · · · + tr(a1 xk−2 + · · · + ak−1 x0 )uk−1 =: B1 u + B2 u2 + · · · + Bk−1 uk−1. So, employing the Gray map yields Φ(Ev(a)) = (Bk−1 , Bk−1 , . . . , Bk−1 + B1 , . . . , Bk−1 + Bk−2 , . . . , Bk−1 + (p − 1)B1 + · · · + (p − 1)Bk−2 )x0 ,...,xk−1 . Since each component of Φ(Ev(a)) contains tr(Bk−1 ), using Lemma 3, it is easy to know that θ(a) = θ(Φ(Ev(a))) = 0 if and only if a ∈ M\{ak−1 uk−1}, where N by the application of Lemma 1. ak−1 ∈ Fpm . Then implies whom (Ev(a)) = p−1 p If a∈M\{ak−1 uk−1}, i.e., a = a′k−1 uk−1 , where a′k−1 ∈ F∗pm , so we get ax = a′k−1 x0 uk−1. Then T r(ax) = tr(a′k−1 x0 )uk−1 =: Cuk−1. Therefore we get Φ(Ev(a)) = (C, C, . . . , C )x0 ,x1 ,...,xk−1 . | {z } pk−1

7

This gives θ(a) = Θ(Φ(Ev(a))) = pk−1

X

x0 ∈Q

= p

k−1 (k−1)m

p

X

X

ωC

x1 ,...,xk−1 ∈Fpm

C

ω .

x0 ∈Q

P After a variable transformation, we see that the term x0 ∈Q ω C equals to Q or N depending on a′k−1 ∈ Q or a′k−1 ∈ N . Due to m is even and p ≡ 1 (mod 4), so G(η) = m m m −p 2 , Q = (−p 2 − 1)/2, and N = (p 2 − 1)/2. Then the statement follows from Lemma 1, i.e., if a′k−1 ∈ Q, whom (Ev(a)) = p−1 (N − p(k−1)(m+1) Q) or whom (Ev(a)) = p p−1 (N − p(k−1)(m+1) N ) when a′k−1 ∈ N . p (c) Let a = a0 + a1 u + · · · + ak−1 uk−1 ∈ R∗ , by a direct calculation we have T r(ax) = tr(a0 x0 ) + tr(a0 x1 + a1 x0 )u + · · · + tr(a0 xk−1 + a1 xk−2 + · · · + ak−1 x0 )uk−1 =: D0 + D1 u + · · · + Dk−1 uk−1 . Since

P

xk−1 ∈Fpm

ω a0 xk−1 = 0, so we can easily check that X

X

ω Dk−1 = 0.

x0 ∈Q x1 ,x2 ,...,xk−1 ∈Fpm

Each of the Gray image Φ(Ev(a)) components contains a Dk−1, so we can get θ(a) = 0. N. Following Lemma 1, we obtain whom (Ev(a)) = p−1 p Theorem 1 together with Lemma 2 implies Φ(Cu,m,p ) is a p-ary code of length N, dimension km, with three nonzero weights ω1 , ω2 and ω3 (ω1 < ω2 < ω3 ) of values m p−1 (N − (p(k−1)(m+1) (p 2 − 1))/2), p p−1 = N, p m p−1 (N + (p(k−1)(m+1) (p 2 + 1))/2), = p

ω1 = ω2 ω3

with respective frequencies f1 , f2 , f3 given by f1 =

pm − 1 pm − 1 , f2 = pkm − pm , f3 = . 2 2 8

m

Remark 1. In the case of p ≡ 3 (mod 4), we know G(η) = p 2 when m is singlym even, and G(η) = −p 2 when m is doubly-even. We also can obtain a p-ary linear code with three nonzero weights by using a similar approach, we omit the proof here. It is easy to check that whether p ≡ 1 (mod 4) or p ≡ 3 (mod 4) when m is even, the weight distribution of Cu,m,p is the same. We give the weight distribution in the following table. Table I. weight distribution of Cu,m,p Weight Frequency 0 1 m −1 m p−1 p (N − (p(k−1)(m+1) (p 2 − 1))/2) p 2 p−1 km p − pm N p m p−1 pm −1 (N + (p(k−1)(m+1) (p 2 + 1))/2) p 2 Example 1. Let p = 3, m = 4, k = 2. Then we obtain a ternary code of parameters [9720, 8, 5832]. The nonzero weights are 5832, 6480 and 7290, its frequencies are 40, 6561 and 40, respectively.

3.2

m is odd and p ≡ 3 (mod 4)

In this case, we know from Subsection 2.4 that G(η) is imaginary, i.e., ℜ(Q) = ℜ(N) = − 12 . Then, we give the following correlation lemma, it establishes a linkage between θ(sa) and ℜ(θ(a)), and leads us to use the similar method in Theorem 1 to discuss the homogeneous weight distribution of Cu,m,p . P Lemma 4. [17, Lemma 2] If p ≡ 3 (mod 4), then p−1 s=1 θ(sa) = (p − 1)ℜ(θ(a)). Theorem 2. For a ∈ R. Assume m is odd and p ≡ 3 (mod 4). (a) If a = 0, then whom (Ev(a)) = 0; (b) If a ∈ M\{0}, then if a ∈ M\{αuk−1 : α ∈ Fpm }, then whom (Ev(a)) = p−1 N, p p−1 k−1 ∗ if a = βu , where β ∈ Fpm , then whom (Ev(a)) = p (N + (p(k−1)(m+1) )/2); (c) If a ∈ R∗ , then whom (Ev(a)) =

p−1 N. p

Proof. We just give the proof of case (b) here, the rest of the cases are similar in Theorem 1. As for case (b), we have ℜ(θ(a)) = 0 when a ∈ M\{αuk−1 : α ∈ Fpm }, 9

(k−1)(m+1)

and ℜ(θ(a)) = − p 2 4.1 and 4.4, we have

when a = βuk−1, where β ∈ F∗pm . Then combining Lemmas

pwhom (Ev(a)) = (p − 1)N − (p − 1)ℜ(θ(a)). The result follows. Thus, combining with Theorem 2 and Lemma 2, we obtain a family of p-ary two-weight codes of parameters [N, km], with two nonzero weights w1′ < w2′ given by w1′ =

p−1 p−1 N, w2′ = (N + (p(k−1)(m+1) )/2), p p

with respective frequencies f1′ , f2′ given by f1′ = pkm − pm , f2′ = pm − 1. Example 2. Let p = 3, m = 3, k = 2. Then we obtain a ternary code of parameters [1053, 6, 702]. The nonzero weights are 702 and 729, its frequencies are 702 and 26, respectively. Remark 2. It is necessary to distinguish the difference between the case when k = 2 in the present paper and the case in [17]. Although the ring and the defining set L are the same as [17], it is not the special case in this paper, because of the different Gray map. Here, we just give an example about three-weight case to show the difference. For instance, if (p, m) = (3, 2), in [17] we obtain a ternary code of parameters [72, 4, 24], and the nonzero weights are {24, 48, 60}, of frequencies 4, 72 and 4, respectively. In the present paper, we obtain a ternary code of parameters [108, 4, 54], and the nonzero weights are {54, 72, 108}, of frequencies 4, 72 and 4, respectively. It is easy to see that the corresponding dimension and frequency are the same, however, the length and the weights of the code are different. In the next section, we will show that they have different dual distance.

4

Further research

In the above section, we have known that the code Cu,m,p is a two-weight code or three-weight code depending on the choice of different conditions. Now we want to research other properties about Cu,m,p . So this section we will study two properties of the code Cu,m,p , one is the optimality, and the other is the dual homogeneous distance. 10

4.1

Optimality

If C is a linear code with parameters [n, k, d], and there is no [n, k, d + 1] code exists, then we call the code C is optimal. The next lemma introduces the Griesmer bound, which applies specifically to the linear codes. Lemma 5. [10, Griesmer bound] Let C be a q-ary code of parameters [n, K, d], where K ≥ 1. Then K−1 Xd ≤ n. qi i=0 Theorem 3. Assume m is odd, and p ≡ 3 (mod 4). If the code Cu,m,p is defined aboven forj given length k and o dimension, then Φ(Cu,m,p ) is optimal if and only if m ≥ pk−1 −2k+1 max k, +1 . 2(k−1) Proof. In the light of Theorem 2, we know Φ(Cu,m,p ) is a [N,lkm,md] code, where PK−1 d+1 N. Next we explore the condition such that > N by using d = ω1′ = p−1 i=0 p pi the the Griesmer bound. Before we start, we guarantee that (m+1)k−m−1 l m≤ km−1, . i.e., m > k. Then we classify the scope of i to determine the value of d+1 pi • If 0 6 i 6 (m + 1)k − m − 2, then

l

d+1 pi

m

• If (m + 1)k − m − 1 6 i 6 km − 1, then

= l

(p−1)(pm −1) (m+1)k−m−2−i p 2

d+1 pi

m

=

p−1 2

+ 1;

· p(m+1)k−2−i .

This implies that K−1 Xl i=0

d + 1m = pi

(m+1)k−m−2 l

X i=0

d + 1m + pi

(p − 1)(pm − 1) = 2 p−1 2 =

km−1 X

km−1 X

i=(m+1)k−m−1

ld + 1m pi

(m+1)k−m−2

X

(p(m+1)k−m−2−i + 1) +

i=0

p(m+1)k−2−i

i=(m+1)k−m−1

pm − 1 (k−1)(m+1) pm − pk−1 (p − 1) + (m + 1)k − m − 1 + . 2 2

P pk−1 1 > 0, and thus we get So we need K−1 ⌈ d+1 i ⌉ − N > 0, i.e., km + k − m − 2 − i=0 p 2 k k n j k−1 o j k−1 −2k+1 −2k+1 + 1. In all, we have m ≥ max k, p 2(k−1) +1 . m ≥ p 2(k−1) 11

Note that the code we considered in Example 2, m = 3 meets the condition in Theorem 3, so the the codes Cu,3,3 is optimal.

4.2

The dual homogeneous distance of Cu,m,p

If x = (x1 , x2 , . . . , xn ) and y = (y1 , y2 , . . . , yn ) are two elements of Rn , their stanP dard inner product is defined by hx, yi = ni=1 xi yi , where the operation is performed ⊥ ⊥ in R. The dual code of Cu,m,p is denoted by Cu,m,p and defined as Cu,m,p = {y ∈ |L| R |hx, yi = 0, ∀x ∈ Cu,m,p }. In this subsection, we compute the dual homogeneous distance of Cu,m,p . A property of the trace function that we need is that it is nondegenerate. The following lemma plays an important role in determining the dual homogeneous distance of Cu,m,p . The process of proof is similarly to [17, Lemma 3], we omit the details. Lemma 6. For a fixed element x ∈ R. If for all a ∈ R, we have that T r(ax) = 0, then x = 0. Theorem 4. For all m ≥ 2, the dual homogeneous distance d′hom of Cu,m,p is 2(p − 1)pk−2 . ⊥ Proof. First, we need to show that Cu,m,p does not contains a codeword that only has a word of homogeneous weight (p−1)pk−2 . If not, we assume that there is a codeword ⊥ that has a word γ = γ0 + γ1 u + · · · + γk−1 uk−1 ∈ R\(uk−1) at some x ∈ L, of Cu,m,p so we know that γ at least exist a coefficient γj 6= 0, where j = 0, 1, . . . , k − 2. Let a = a0 + a1 u + · · · + ak−1uk−1 ∈ R, x = x0 + x1 u + · · · + xk−1 uk−1 ∈ L. Then we have γT r(ax) = 0, which gives k equations, according to the first equation and Lemma 6, we have tr(γ0a0 x0 ) = 0, γ0 x0 = 0, but x0 6= 0, so γ0 = 0. Applying the same technique to the next equations, we get γ1 = γ2 = · · · = γk−2 = 0, a contradiction. ⊥ Next, we prove that there exists a codeword of Cu,m,p that has homogeneous weight k−2 k−2 k−1 ⊥ does not 2(p − 1)p . Since 2(p − 1)p > p , so we need to show that Cu,m,p k−1 contain a codeword that only has a word of homogeneous weight p . We can use a similar approach as above to prove it, and omit here. Then we assume that there ⊥ exists a codeword of Cu,m,p have two values α = α0 + α1 u + · · · + αk−1 uk−1 , β = β0 + β1 u + · · · + βk−1 uk−1 ∈ R\(uk−1) at some x, y ∈ L, where x = x0 + x1 u + · · · + xk−1 uk−1 , y = y0 + y1 u + · · · + yk−1uk−1 ∈ L. Then we have αT r(ax) + βT r(ay) = 0, i.e., k equations as follows:

12

  α0 x0 + β0 y0 = 0;     α0 x1 + α1 x0 + β0 y1 + β1 y0 = 0; ..  .     αx 0 k−1 + α1 xk−2 + · · · + αk−1 x0 + β0 yk−1 + β1 yk−2 + · · · + βk−1 y0 = 0.

According to the above equations, we can treat it as a system of homogeneous linear equations with unknowns αi , βj , where i, j = 0, 1, . . . , k − 1. It is clear to know this system has nonzero solutions. Due to x0 , y0 ∈ Q, without loss of generality, we let −1 α0 = x−1 0 6= 0, β0 = −y0 6= 0, so such α, β exist, and the assumption is correct.

5

Application of the linear codes to secret sharing schemes

5.1

The covering problem of linear codes

The support of a vector c = (c0 , c1 , . . . , cn−1 ) ∈ Fnq is defined as {0 ≤ i ≤ n−1|ci 6= 0}. We say that a vector x covers a vector y if the support of x contains the support of y. A minimal codeword of a linear code C is a nonzero codeword that does not cover any other nonzero codeword. The covering problem of a linear code is to determine all the minimal codewords. However, in general determining the minimal codewords of a given linear code is a difficult task. In special cases, the Ashikhmin-Barg lemma [1] is very useful in determining the minimal codewords. Lemma 7. (Ashikhmin-Barg) In an [n, k; q] code C, let wmin and wmax be the minimum and maximum nonzero weights, respectively. If q−1 wmin > , wmax q then all nonzero codewords of C are minimal. We can infer from there the support structure for the codes of this paper. Proposition 2. If one of the following two conditions satisfied (1) m ≥ 4 even; (2) m > 3 odd, and p ≡ 3 (mod 4), 13

(1)

then all the nonzero codewords of Φ(Cu,m,p ) are minimal. Proof. Following Theorem 1 and Lemma 7, we know wmin = w1 , and wmax = w3 . Then we calculate pw1 − (p − 1)w3 as follows: m

pw1 − (p − 1)w3

 m p − 1 (k−1)(m+1)  pm − 1 p 2 + 1 p + − p 2 +1 . = p 2 2 m

m

Since p is odd prime and m ≥ 4 even, so pm + p 2 − 2p 2 +1 > 0. The inequality in (1) is satisfied. Likewise, according to Theorem 2 and Lemma 7, we know wmin = w1′ , and wmax = w2′ . Then we calculate pw1′ − (p − 1)w2′ as follows: p − 1 (k−1)(m+1)  pm − 1 p − 1  ′ ′ . p − pw1 − (p − 1)w2 = p 2 2 Since for any odd prime p ≡ 3 (mod 4) and m > 3 odd, we have pm − p > 0. Hence the inequality in (1) is satisfied.

5.2

Secret sharing schemes

Secret sharing schemes (SSS) were first introduced by Blakly [3] and Shamir [14] at the end of 70s twentieth century. Since then, many constructions have been proposed. Massey’s scheme is a construction of such a scheme which pointed out the relationship between the access structure and the minimal codewords of the dual code of the underlying code [13]. See [19] for a detailed explanation of the mechanism of that scheme. It would be interesting to know the dual Hamming distance (not only the dual homogeneous distance), as this would impact the SSS democratic or dictatorial character [7]. We leave this as an open problem to the diligent reader.

6

Conclusion

The objective of the present paper is to study a family of trace codes over a finite ring. Using a character sum approach, we have been able to determine their homogeneous weight distribution, then a class of abelian p-ary two-weight codes and three-weight codes are obtained by the application of a linear Gray map. Furthermore, we have determined their dual homogeneous distance. In particular, we have proved that the code Φ(Cu,m,p ) is optimal under some condition when p ≡ 3 (mod 4) and m 14

odd. Determining the dual Hamming distance of the considered codes is a challenging open problem.

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[15] Shi, M., Liu, Y., Sol´e, P.: Optimal two weight codes over F2 + uF2 , IEEE Communication Letters, Accepted with doi: 10.1109/LCOMM.2016.2614934. [16] Shi, M., Liu, Y., Sol´e, P.: Optimal two weight codes from trace codes over a non-chain ring, Discrete Applied Mathematics, (Accepted). [17] Shi, M., Wu, R., Liu, Y., Sol´e, P.: Two and three weight codes over Fp + uFp , Cryptography & Communications, 2016, doi:10.1007/s12095-016-0206-5 [18] Shi, M., Zhu, S., Yang, S.: A class of optimal p-ary codes from one-weight codes over Fp [u]/(um), Journal of the Franklin Institute, 2013, 350(5):929-937. [19] Yuan, J., Ding, C.: Secret sharing schemes from three classes of linear codes, IEEE Transactions on Information Theory, 2006, 52(1):206-212.

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