Quadratic Residue Codes over Fp+vFp+v2Fp

Quadratic Residue Codes over Fp + vFp + v 2 Fp Yan Liu1 , Minjia Shi1(B) , and Patrick Sol´e2 1

2

School of Mathematical Sciences, Anhui University, Hefei, China [email protected] CNRS/LTCI Telecom Paris Tech, 46 rue Barrault, 75013 Paris, France

Abstract. This article studies quadratic residue codes of prime length q over the ring R = Fp + vFp + v 2 Fp , where p, q are distinct odd primes. After studying the structure of cyclic codes of length n over R, quadratic residue codes over R are defined by their generating idempotents and their extension codes are discussed. Examples of codes and idempotents for small values of p and q are given. As a by-product almost MDS codes over F7 and F13 are constructed. Keywords: Cyclic codes potents · Dual codes

· Quadratic residue codes · Generating idem-

MSC (2010): Primary 94B15; Secondary 11A15.

1

Introduction

Cyclic codes form an important subclass of linear block codes, studied from the fifties onward. Their clean algebraic structure as ideals of a quotient ring of a polynomial ring makes for an easy encoding. Quadratic residue codes are a special kind of cyclic codes of prime length introduced to construct self-dual codes by adding an overall parity-check. Quadratic residue codes over finite fields have been studied extensively by Assmus and Mattson in a series of research reports [1]. Since then, coding theorists have studied quadratic residue codes and their properties over rings that are not fields. Cyclic codes and quadratic residue codes over Z4 were studied in [2]. Tan X investigated a family of quadratic residue codes over Z2m in [3]. The above rings are local. In [4], Kaya A et al. studied quadratic residue codes over Fp + vFp and their Gray images. In [5], Zhang T et al. used another way to prove that some properties of quadratic residue codes over Fp + vFp . Cyclic codes and the weight enumerators of linear codes over F2 + vF2 + v 2 F2 were given in [6]. It is worth mentioning that Kaya A et al. have done great contribution to new extremal and optimal binary self-dual codes from quadratic residue codes over F2 + uF2 + u2 F2 in [7]. This research is supported by NNSF of China (61202068), Talents youth Fund of Anhui Province Universities (2012SQRL020ZD). Technology Foundation for Selected Overseas Chinese Scholar, Ministry of Personnel of China (05015133) and the Project of Graduate Academic Innovation of Anhui University (N0. yfc100005). c Springer International Publishing Switzerland 2015  C ¸ . Ko¸ c et al. (Eds.): WAIFI 2014, LNCS 9061, pp. 204–211, 2015. DOI: 10.1007/978-3-319-16277-5 12

Quadratic Residue Codes over Fp + vFp + v 2 Fp

205

Following the above trend, this paper is devoted to studying quadratic residue codes over the non-local ring Fp + vFp + v 2 Fp , where p is an odd prime, thus generalizing [7]. First, we introduce the structure of cyclic codes over Fp + vFp + v 2 Fp , and prove that there is only one idempotent generator for each code. Next, we define the quadratic residue codes over R, and derive a closed form expression for their idempotents. The material is organized as follows. The next section contains the basics of codes over rings that we need for further notice. Section 3 derives the structure of cyclic codes over R. Section 4 introduces quadratic residue codes, and Sect. 5 studies their extensions. Section 6 contains a numerical example and Sect. 7 draws conclusions and suggests some open problems.

2

Preliminary Results

Throughout, we let R denote the commutative ring Fp + vFp + v 2 Fp , where v 3 = v, where p is an odd prime. R is a characteristic p ring of size p3 . Denote by η1 , η2 , η3 respectively the following elements of R. η1 = 2−1 v + 2−1 v 2 , η2 = −2−1 v + 2−1 v 2 , η3 = 1 − v 2 . 3 A direct calculation shows that ηi2 = ηi , ηi ηj = 0, i=1 ηi = 1, where i, j = 1, 2, 3 and i = j. The decomposition theorem of ring theory tells us that R = η1 R ⊕ η2 R ⊕ η3 R. A code C of length n over R is an R-submodule of Rn . An element of C is called a codeword of C. A generator matrix of C is a matrix whose rows generate C. The Hamming weight of a codeword is the number of non-zero components. Let x = (x1 , x2 , · · · , xn ) and y = (y1 , y2 , · · · , y n ) be two n elements of Rn . The Euclidean inner product is given as (x, y) = i=1 xi yi . ⊥ The dual code C of C with respect to the Euclidean inner product is defined as C ⊥ = {x ∈ Rn |(x, y) = 0, for ∀ y ∈ Rn }. C is self-dual if C = C ⊥ , C is self-orthogonal if C ⊆ C ⊥ . In the sequel we let Rq := R[x]/(xq − 1). A polynomial f (x) is abbreviated as f if there is no confusion. An idempotent is an element e ∈ Rq such that e2 = e. We characterize all idempotents in Rq in the following lemma. Lemma 2.1. η1 f1 + η2 f2 + η3 f3 is an idempotent in Rq if and only if fi are idempotents in Fp [x]/(xq − 1), where i = 1, 2, 3. Proof. Let g = η1 f1 + η2 f2 + η3 f3 be an idempotent in Rq , then we have [η1 f1 + η2 f2 + η3 f3 ]2 = η1 f12 + η2 f22 + η3 f32 = η1 f1 + η2 f2 + η3 f3 , which implies fi2 = fi , where i = 1, 2, 3. Conversely, if fi are idempotents in Fp [x]/(xq − 1), then η1 f1 + η2 f2 + η3 f3 is an idempotent in Rq , since [η1 f1 + η2 f2 + η3 f3 ]2 = η12 f12 + η22 f22 + η32 f32 = η1 f1 + η2 f2 + η3 f3 . For all a ∈ F∗p , the map μa : Fp → Fp is defined by μa (i) = ai (mod p) and   it acts on polynomials as μa ( i hi xi ) = i hi xμa (i) , hi denote coefficients. It is easily observed that μa (f g) = μa (f )μa (g) for polynomials f and g in Rq .

206

Y. Liu et al.

Let S be a commutative ring with identity, according to the results in [2], then we have the following results. Theorem 2.2. Let C and D be cyclic codes of length n over S generated by the idempotents f1 , f2 in S[x]/(xn − 1), then C ∩ D and C + D are generated by the idempotents f1 f2 and f1 + f2 − f1 f2 , respectively. Theorem 2.3. Let C be a cyclic code of length n over S generated by the idempotent f in S[x]/(xn − 1), then its dual C ⊥ is generated by the idempotent 1 − f (x−1 ). Definition 2.4. For any r ∈ R, r can be uniquely expressed as r = η1 a + η2 b + η3 c , where a , b , c ∈ Fp , the Gray map Φ from R to F3p is defined as Φ(η1 a + η2 b + η3 c ) = (a + c , a + b , b + c ), which can be generalized to Rn to F3n p naturally.

3

Cyclic Codes over Fp + vFp + v 2 Fp

In this section, we recall some results on cyclic codes over R, in particular that there is only one idempotent e ∈ Rq such that C = (e). If A, B are codes over R, we write A⊕B to denote the code {a+b|a ∈ A, b ∈ B}. Let C be a linear code of length n over R, we define C1 = {a ∈ Fnp |∃b, c ∈ Fnp |η1 a + η2 b + η3 c ∈ C}, C2 = {b ∈ Fnp |∃a, c ∈ Fnp |η1 a + η2 b + η3 c ∈ C}, C3 = {c ∈ Fnp |∃a, b ∈ Fnp |η1 a + η2 b + η3 c ∈ C}, then Ci (i = 1, 2, 3) are linear codes of length n over Fp , C = η1 C1 ⊕ η2 C2 ⊕ η3 C3 and |C| = |C1 ||C2 ||C3 |. The following two theorems can be found in [8], we give the proof for copmpleteness. Theorem 3.1. [8] Let C = η1 C1 ⊕ η2 C2 ⊕ η3 C3 be a cyclic code of length n over R, then C = (η1f1 , η2 f2 , η3 f3 ), where Ci = (fi ), fi ∈ Fp [x](i = 1, 2, 3), fi |xn − 1, 3  and |C| = p3n− i=1 deg(fi ) . Proof. First, we prove that if C = η1 C1 ⊕ η2 C2 ⊕ η3 C3 is a cyclic code over R if and only if Ci (i = 1, 2, 3) are cyclic codes over Fp . For any (α0 , α1 , . . . , αn−1 ) ∈ C, where αi = η1 ai + η2 bi + η3 ci , i = 0, 1, · · · , n − 1. Let a = (a0 , a1 , . . . , an−1 ) ∈ C1 , b = (b0 , b1 , . . . , bn−1 ) ∈ C2 , c = (c0 , c1 , . . . , cn−1 ) ∈ C3 . T is shift operator. If C = η1 C1 ⊕ η2 C2 ⊕ η3 C3 is a cyclic code over R, then T (α) = η1 T (a) + η2 T (b) + η3 T (c) ∈ C. Hence T (a) ∈ C1 , T (b) ∈ C2 and T (c) ∈ C3 . On the other hand, if Ci (i = 1, 2, 3) are cyclic codes over Fp , then T (a) ∈ C1 , T (b) ∈ C2 , T (c) ∈ C3 . Hence T (α) = η1 T (a) + η2 T (b) + η3 T (c) ∈ C, which means that C is a cyclic code over R. Next, assuming that Ci = (fi ), where fi ∈ Fp [x], fi |xn − 1, |Ci | =  pn−deg(fi ) , (i = 1, 2, 3). It is obvious C ⊆ (η1 f1 , η2 f2 , η3 f3 ). Now, let r = η1 f1 r1 + η2 f2 r2 + η3 f3 r3 ⊆ (η1 f1 , η2 f2 , η3 f3 ), where r, ri ∈ R[x], i = 1, 2, 3, so there exit ai , bi , ci ∈ Fp [x] such that ri = η1 ai + η2 bi + η3 ci . Hence r = η1 f1 r1 + η2 f2 r2 + η3 f3 r3 = η1 f1 a1 + η2 f2 b2 + η3 f3 c3 ∈ C. That is (η1 f1 , η2 f2 , η3 f3 ) ⊆ C. Theorem 3.2. [8] Let C be a cyclic code of length n over R, then there exists a polynomial g ∈ R[x] such that C = (g) and g|xn − 1.

Quadratic Residue Codes over Fp + vFp + v 2 Fp

207

Proof. By Theorem 3.1, we may assume that C = (η1 f1 , η2 f2 , η3 f3 ), where Ci = (fi )(i = 1, 2, 3). Let g = η1 f1 + η2 f2 + η3 f3 . Clearly, (g) ⊆ C. On the other hand, ηi fi = ηi g, i = 1, 2, 3. This gives C ⊆ (g), hence C = (g). According to Theorems 3.2, we have the following theorem, the proof of which follows by the Bezout identity like in the field alphabet case. Theorem 3.3. Let C = η1 C1 ⊕ η2 C2 ⊕ η3 C3 be a cyclic code of length n over R, where (n, p) = 1, Ci = (fi ), fi (i = 1, 2, 3) are idempotents, then there is only one idempotent e ∈ C such that C = (e), where e = η1 f1 + η2 f2 + η3 f3 .

4

Quadratic Residue Codes over Fp + vFp + v 2 Fp

In this section, quadratic residue codes over R are defined in terms of their idempotent generators. Let q be an odd prime such that q ≡ ±1(mod 4) and non-residues modulo q, let Qq and Nq be the sets of quadratic residues and  respectively. We use the notations g1 = i∈Qq xi , g2 = i∈Nq xi and h denotes the polynomial corresponding to the all one vector of length q, i.e. h = 1+g1 +g2 . Following classical notation [1] we let for i = 1, 2 the codes Ci (resp. Ci ) be generated by gi (resp. (x + 1)gi ). Lemma 4.1. [1] If p > 2 and q = 4k±1, then idempotent generators of quadratic residue codes C1 , C1 , C2 , C2 over Fp are Eq (x) = 12 (1 + 1q ) + 12 ( 1q − θ1 )g1 + 12 ( 1q + 1 1 1 1 1 1 1 1 1 1 1 1 1 θ )g2 , Fq (x) = 2 (1 − q ) − 2 ( q + θ )g1 − 2 ( q − θ )g2 , En (x) = 2 (1 + q ) + 2 ( q − 1 1 1 1 1 1 1 1 1 1 1 1 θ )g2 + 2 ( q + θ )g1 , Fn (x) = 2 (1 − q ) − 2 ( q + θ )g2 − 2 ( q − θ )g1 , respectively, where θ denotes Gaussian sum, χ(i) denotes Legendre symbol, that is, ⎧ ⎪ p|i; q−1 ⎨0,  i θ= χ(i)α , χ(i) = 1, i ∈ Qq ; ⎪ ⎩ i=1 −1, i ∈ Nq , where α is a primitive q th root of unit over some extension field of Fp . For convenience, we set e1 = Eq (x), e1 = Fq (x), e2 = En (x), e2 = Fn (x). Using Lemmas 2.1 and 4.1, we can obtain the following theorem, which plays an important role in the main results. Lemma 4.2. η1 ei + η2 ej + η3 ek , η1 ei + η2 ej + η3 ek are idempotents in Rq , where ei , ej , ek are not all equal, ei , ej , ek are not all equal (i, j, k = 1, 2). Next, define the quadratic residue codes over R in terms of their idempotent generators. Definition 4.3. Let q be an odd prime such that p is a quadratic residue modulo q. Set Q3 = (η1 e2 + η2 e1 + η3 e1 ), Q2 = (η1 e1 + η2 e2 + η3 e1 ), Q3 = (η1 e1 + η2 e1 + η3 e2 ), Q4 = (η1 e1 + η2 e2 + η3 e2 ), Q5 = (η1 e2 + η2 e1 + η3 e2 ), Q6 = (η1 e2 + η2 e2 + η3 e1 ), S1 = (η1 e2 + η2 e1 + η3 e1 ), S2 = (η1 e1 + η2 e2 + η3 e1 ), S3 = (η1 e1 + η2 e1 + η3 e2 ), S4 = (η1 e1 + η2 e2 + η3 e2 ), S5 = (η1 e2 + η2 e1 + η3 e2 ), S6 = (η1 e2 + η2 e2 + η3 e1 ). These twelve codes are called quadratic residue codes over R of length q.

208

Y. Liu et al.

3 Note that i=1 ηi = 1, we have Qi = ((1 − ηi )e1 + ηi e2 ), Q3+i = ((1 − ηi )e2 + ηi e1 ), Si = ((1 − ηi )e1 + ηi e2 ), S3+i = ((1 − ηi )e2 + ηi e1 )(i = 1, 2, 3). As in the case of quadratic residue codes over finite ring F2 + uF2 + u2 F2 , the properties of quadratic residue codes over R differ from the cases q ≡ 3 (mod 4) and q ≡ 1 (mod 4). Theorem 4.4. If q ≡ 3 (mod 4), with the notation as in Definition 4.3, the following assertions hold for quadratic residue codes over R: (a) Qi and Si are equivalent to Q3+i and S3+i , respectively, i = 1, 2, 3; (b) Qi ∩ Q3+i = ( 1q h), Qi + Q3+i = Rq , i = 1, 2, 3; (c) Qj = Sj + ( 1q h), j = 1, 2, 3, 4, 5, 6; (d) |Qj | = (p3 )(q+1)/2 , |Sj | = (p3 )(q−1)/2 , j = 1, 2, 3, 4, 5, 6; (e) Sj are self-orthogonal and Q⊥ j = Sj , j = 1, 2, 3, 4, 5, 6; (f ) Si ∩ S3+i = {0}, Si + S3+i = (1 − 1q h), i = 1, 2, 3. Proof. (a) Let n ∈ Nq , then μn (g1 ) = g2 and μn (g2 ) = g1 . Hence μn (e1 ) = e2 , μn (e2 ) = e1 , μn (e1 ) = e2 , μn (e2 ) = e1 . Therefore, μn ((1 − ηi )e1 + ηi e2 ) = (1 − ηi )e2 + ηi e1 , μn ((1 − ηi )e1 + ηi e2 ) = (1 − ηi )e2 + ηi e1 , which implies Qi and Q3+i are equivalent, Si and S3+i are equivalent. (b) By direct calculation, we have ((1 − ηi )e1 + ηi e2 ) · ((1 − ηi )e2 + ηi e1 ) = ((1 − ηi )e1 + ηi e2 ) · [((1 − ηi )e1 + ηi e2 ) + ((1 − ηi )e2 + ηi e1 ) − 1] = ((1 − ηi )e1 + ηi e2 ) · (e1 + e2 − 1) = ((1 − ηi )e1 + ηi e2 ) · 1q h = 1q · {(1 − ηi )[ 12 (1 + 1q )h + 12 ( 1q − 1 q−1 1 1 1 q−1 1 1 1 1 1 q−1 1 1 1 q−1 1 θ ) 2 h + 2 ( q + θ ) 2 h] + ηi [ 2 (1 + q )h + 2 ( q + θ ) 2 h + 2 ( q − θ ) 2 h]} = q h. According to Theorem 2.2, we have Qi ∩ Q3+i = ( 1q h). ((1 − ηi )e1 + ηi e2 ) + ((1 − ηi )e2 + ηi e1 ) + ((1 − ηi )e1 + ηi e2 ) · ((1 − ηi )e2 + ηi e1 ) = e1 + e2 − 1q h = 1. Hence Qi + Q3+i = Rq . (c) By direct calculation, we have ((1 − ηi )e1 + ηi e2 ) · 1q h = 1q · {(1 − ηi )[ 12 (1 − 1 1 1 1 q−1 1 1 1 q−1 1 1 1 1 1 q−1 1 1 q )h − 2 ( q + θ ) 2 h − 2 ( q − θ ) 2 h] + ηi [ 2 (1 − q )h − 2 ( q − θ ) 2 h − 2 ( q + 1 q−1 1 1   θ ) 2 h]} = 0 = ((1 − ηi )e2 + ηi e1 ) · q h. Therefore, Sj ∩ ( q h) = 0. Since ((1 − 1     ηi )e1 + ηi e2 ) + q h = ((1 − ηi )e1 + ηi e2 ) + ((1 − ηi ) + ηi ) 1q h = ((1 − ηi )e1 + ηi e2 ).((1 − ηi )e2 + ηi e1 ) + 1q h = ((1 − ηi )e2 + ηi e1 ) Hence, Qj = Sj + ( 1q h). (d) According to (a) and (b), we have |Qi ∩ Q3+i | = |( 1q h)| = p3 . Since |Qi ||Q3+i | |Qi ∩Q3+i | |Sj +( 1q h)|

(p3 )q = |Qi +Q3+i | =

=

|Qi |2 p3 . Thus, |Sj ||( 1q h)| =

|Qi | = |Q3+i | = (p3 )(q+1)/2 . Since

= |Qj | = = p3 |Sj |, we have |Sj | = (p3 )(q−1)/2 . (p ) (e) Since −1 ∈ Nq , according to Theorem 2.3, the generating idempotent of C1⊥ of C1 = (e1 ) is 1 − e1 (x−1 ) = 1 − 12 (1 + 1q ) − 12 ( 1q − θ1 )g2 − 12 ( 1q + θ1 )g1 = 1 1 1 1 1 1 1 1  2 (1 − q ) − 2 ( q + θ )g1 − 2 ( q − θ )g2 = e1 . Similarly, the generating idempotent of C2⊥ is e2 . Using Theorem 2.3, the generating idempotent of Q⊥ i (i = 1, 2, 3) are (1 − ηi )e1 + ηi e2 , which implies Q⊥ i = Si . According to c), we have Si ⊆ Qi = Si⊥ . Hence, Si (i = 1, 2, 3) are self-orthogonal. Similarly, Q⊥ 3+i = S3+i and S3+i (i = 1, 2, 3) are self-orthogonal. (f ) Since ((1 − ηi )e1 + ηi e2 ) · ((1 − ηi )e2 + ηi e1 ) = ((1 − ηi )e1 + ηi e2 ) · [((1 − ηi )e1 + ηi e2 ) + ((1 − ηi )e2 + ηi e1 ) − 1] = ((1 − ηi )e1 + ηi e2 ) · (e1 + e2 − 1) = 3 (q+1)/2

Quadratic Residue Codes over Fp + vFp + v 2 Fp

209

((1 − ηi )e1 + ηi e2 ) · (− 1q h) = 0. That is, Si ∩ S3+i = 0. ((1 − ηi )e1 + ηi e2 ) + ((1 − ηi )e2 + ηi e1 ) = e1 + e2 = 1 − 1q h. Hence, Si + S3+i = (1 − 1q h). Similar to the proof of Theorem 4.4, we have the following theorem. Theorem 4.5. If q ≡ 1 (mod 4), with the notation as in Definition 4.3, then the following assertions hold for quadratic residue codes over R: (a) Qi and Si are equivalent to Q3+i and S3+i , respectively, i = 1, 2, 3; (b) Qi ∩ Q3+i = ( 1q h), Qi + Q3+i = Rq , i = 1, 2, 3; (c) Qj = Sj + ( 1q h), j = 1, 2, 3, 4, 5, 6; (d) |Qj | = (p3 )(q+1)/2 , |Sj | = (p3 )(q−1)/2 , j = 1, 2, 3, 4, 5, 6; ⊥ (e) Q⊥ i = S3+i , Q3+i = Si , i = 1, 2, 3; (f ) Si ∩ S3+i = 0, Si + S3+i = (1 − 1q h), i = 1, 2, 3.

5

Extended Quadratic Residue Codes over Fp + vFp + v 2 Fp

In this section, we discuss the properties of extended quadratic residue codes over R. Definition 5.1. The extended code of a code C over R will be denoted by C, which is the code obtained by adding a specific column to the generator matrix  j as of C. In addition, define the generator matrix of Q ∞ 0 ⎜ 0 ⎜ ⎜ . ⎝ .. 1 ⎛

0

1

2

···

q−1

⎟ ⎟ ⎟, ⎠

Gj 1

1

1

⎞

···

1

where Gj generates Sj (j = 1, 2, · · · 6), and the row above the horizontal bar shows the column labelling by Fq ∪ ∞. Theorem 5.2. If q ≡ 3 (mod 4), with the notation Qj (j = 1, 2, · · · 6) as in ⊥  j . In particular, if 1 + 1 = 0, then Qj are self-dual. Definition 4.3, then Q = Q j

q

Proof. Theorem 4.4 tells us that Qj = Sj + ( 1q h)(j = 1, 2, 3, 4, 5, 6), then the generator matrix of Qj are ∞ 0 ⎜ 0 ⎜ ⎜ .. ⎝ . −1 ⎛

0

1

2

···

q−1

Gj 1 q

1 q

1 q

⎞ ⎟ ⎟ ⎟, ⎠

···

1 q

210

Y. Liu et al.

where Gj are a generator matrix of Sj . Since Sj are self-orthogonal, any two rows of Gj are orthogonal. According to the proof of (c) in Theorem 4.4, we know that each line of Gj are orthogonal together with the vector ( 1q h). Since ⊥  j | = (p3 )(q+1)/2 . That is, Q⊥ = Q  j . In (1, h) · (−1, 1 h) = 0, then |Q | = |Q j

q

particular, if 1 +

1 q

j

= 0, Qj are linear codes generated by the matrix

∞ 0 ⎜ 0 ⎜ Gj = ⎜ . ⎝ .. −1

0

⎛

1

2

···

q−1

⎟ ⎟ ⎟, ⎠

Gj −1 −1 −1

⎞

···

−1

⊥

Obviously, (1, h) ∈ Gj . Hence, Qj = Qj . That is, Qj are self-dual. Similar to the proof of Theorem 5.2, we have the following theorem. Theorem 5.3. If q ≡ 1 (mod 4), with the notation Qj (j = 1, 2, · · · 6) as in ⊥ 1  3+i , Q⊥  Definition 4.3, then Qi = Q 3+i = Qi (i = 1, 2, 3). In particular, if 1+ = 0, ⊥

q

⊥

then Qi = Q3+i , Q3+i = Qi (i = 1, 2, 3).

6

Numerical Examples

In this section, we give some examples to validate the main conclusions obtained in this paper. The parameters given are that of the finite field image defined for all ai ∈ Fp by the formula   3 ai ηi = (a1 + a3 , a1 + a2 , a2 + a3 ). Φ i=1

Note that φ does not map self-dual codes to self dual codes, but it does give some good Hamming distance codes. • Let p = 3 and q = 11. The sets of quadratic residues and non-residues modulo q are Qq = {1, 3, 4, 5, 9}, and Nq = {2, 6, 7, 8, 10}, respectively. Thus,  g1 (x) = i∈Qq xi = x + x3 + x4 + x5 + x9 and g2 (x) = i∈Nq xi = x2 + x6 + x7 + x8 + x10 . According to Lemma 4.1, e1 = 1 + g2 , e2 = 1 + g1 , and ei = −gi , where i = 1, 2. By Definition 4.3, Q3 = (−v 2 (x + x3 + x4 + x5 + x9 ) − (1 − v 2 )(x2 + x6 + x7 + x8 + x10 )) and S3 = (v 2 (1 + x2 + x6 + x7 + x8 + x10 ) + (1 − v 2 )(1 + x + x3 + x4 + x5 + x9 )). Theorem 4.4 tells us that S3 is self-orthogonal and Q⊥ 3 = S3 . According to Theorem 5.2, Q3 is self-dual. It image by φ has parameters [36, 18, 11]. • Let q = 3 and p = 7 or p = 13. The code φ(Q1 ) is an almost MDS [9, 6, 3] code in the sense of [9]. It is not nearly MDS since the dual has parameters [9, 3, 3]. • Let q = 23 and p = 3. The code φ(Q1 ) has parameters [72, 36, 13].

Quadratic Residue Codes over Fp + vFp + v 2 Fp

7

211

Conclusion

This article gives the definition and some properties of quadratic residue codes over the ring Fp +vFp +v 2 Fp , subject to the restriction v 3 = v, where p is an odd prime. In [7], the quadratic residue codes over F2 +vF2 +v 2 F2 (v 3 = v) are studied in detail. In this paper, we generalize the structural results of [7] by replacing F2 by Fp , for odd primes p. We derived the generating idempotents of quadratic residue codes. Some Gray images turn out to be almost MDS. Alternative Gray maps compatible with duality and/or cyclicity are worth exploring.

References 1. MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North Holland Publishing Co, Amsterdam (1977) 2. Pless, V., Qian, Z.: Cyclic codes and quadratic residue codes over Z4 . IEEE Trans. Inform. Theory 42(5), 1594–1600 (1996) 3. Tan, X.: A family of quadratic residue codes over Z2m . In: Fourth International Conference on Emerging Intelligent Data Web Technologies, pp. 236–240 (2013) 4. Kaya, A., Yildiz, B., Siap, I.: Quadratic Residue Codes over Fp +vFp and their Gray Images. arXiv preprint arXiv: 1305, 4508 (2013) 5. Zhang, T., Zhu, S.X.: Quadratic residue codes over Fp + vFp . J. Univ. Sci. Technol. China 42(3), 208–213 (2012). In Chinese 6. Shi, M.J., Sol´e, P., Wu, B.: Cyclic codes and the weight enumerator of linear codes over F2 + vF2 + v 2 F2 . Appl. Comput. Math 12(2), 247–255 (2013) 7. Kaya, A., Yildiz, B., Siap, I.: New extremal binary self-dual codes of length 68 from quadratic residue codes over F2 +uF2 +u2 F2 . Finite Fields Appl. 29, 160–177 (2014) 8. Cao, D.C., Zhu, S.X.: Constacyclic codes over the ring Fq + vFq + v 2 Fq . J. Hefei Univ. Technol. 36(12), 1534–1536 (2013). In Chinese 9. de Boer, M.: Almost MDS codes. Des. Codes Crypt. 9(2), 143–155 (1996)