MDR Codes over Zk Steven T. Dougherty Department of Mathematics University of Scranton Scranton, PA 18510 USA AND Keisuke Shiromoto Department of Mathematics Kumamoto University 2-39-1, Kurokami Kumamoto 860-8555 Japan

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Running head: MDR Codes over Zk Name: Steven T. Dougherty and Keisuke Shiromoto Contact Author: Keisuke Shiromoto Address: Department of Mathematics Kumamoto University 2-39-1, Kurokami Kumamoto 860-8555 Japan Telephone: + 81 96 342 3331; Fax: + 81 96 342 3341; E-mail: [email protected]

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Abstract In this paper, we study Maximum Distance with respect to Rank codes over the ring Zk . We generalize the construction of BCH and Reed-Solomon Codes and apply the Generalized Chinese Remainder Theorem to construct codes. Index Terms: Maximum Distance with respect to Rank Codes, Codes over Rings.

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1

Introduction

A code of length n over the ring Zk is a subset of Znk . If the code is also a submodule of Znk , then we say that the code is linear. In particular, if the code is a free submodule of Znk , we say that the code is free. In this paper, we assume all codes are linear unless otherwise P indicated. To the ambient space we attach the natural inner-product, i.e. hv, wi = vi wi and denote the code orthogonal to C under this inner-product as C ⊥ . We shall give the following known results, using the notations of [12]. Any finitely generated submodule of Znk is isomorphic to Zk /f1 Zk ⊕ Zk /f2 Zk ⊕ . . . ⊕ Zk /fn Zk ,

(1)

where fi are positive integers with f1 | f2 | . . . | fn | k. Given a finitely generated submodule of Znk , we define the rank as |{i | fi 6= 1}|, the free rank as |{i | fi = k}| and we note that Q the number of elements in the module is fi . In [12], the following is shown: For a code C over Zk of length n with minimum distance d(C), d(C) ≤ n − rank(C) + 1. Hence we make the following definition: If C is a code over Zk of length n with d(C) = n−rank(C)+1, then we say that C is a Maximum Distance with respect to Rank code (MDR) and if the rank is equal to the free rank then we say that it is a Free Maximum Distance with respect to Rank code (FMDR). An FMDR code can be thought of as an MDS code, see for example [5], [6], [1], and [2]. For MDR codes over Zpm , where p is a prime number, Shiromoto [13] gave a duality and a matrix characterization. If k is a square then a self-dual trivial MDR code exists for length 1. Namely, if k = m2 , then m generates a self-dual code with minimum weight 1 and rank 1, hence n−rank(C)+1 = 1 − 1 + 1 = 1 and the code is MDR. This situation does not occur for fields. However, like fields trivial MDR [n, 1, n] codes are generated by the all one-vector, trivial MDR [n, n−1, 2] codes are taken by the orthogonal to this code, and trivial MDR codes [n, n, 1] codes are Znk . Note that here [n, r, d] codes refer to codes of length n, rank r and minimum Hamming weight d. However, other trivial MDR codes exist with these parameters, for example the code generated by the all-2 vector in Zn4 .

1.1

Bounds

In [7], an upper bound is given on the maximum number of vectors with a minimum Hamming weight of a not necessarily linear code. This result is also found in [3] and [6]. It is stated in terms of finite fields, however its proof applies to Zk as well. Namely, given h vectors of minimum weight d, they are added to all k d−1 possible vectors with the last n − d + 1 coordinates equal to 0. These sums are distinct and hk d−1 in number. This must be bounded above by the cardinality of the ambient space and so hk d−1 ≤ k n giving the following result. 4

Theorem 1.1 The number of code words of length n of a code over Zk with minimum Hamming distance d is at most k n−d+1 . Note that this result coincides with the bound for a FMDR code. This bound is referred to as the Joshi bound, [7], [3]. In [6], this bound is proved for (not necessary linear) codes over arbitrary alphabets. Moreover there is another theorem in [3] which can be easily generalized. Theorem 1.2 When n ≤ k + 1 and d = n − 1 or when n ≤ k and d = n, the Joshi bound can always be attained if k is a number for which k − 1 mutually orthogonal Latin squares exist. In fact, the example of this theorem, given in [3], is over Z4 even though it is phrased over the finite field of order 4. Moreover many other results in this chapter are applicable. In particular, the following result (Theorem 10.1.5, [3]) is related to MDR codes. If {Ai | Ai = (ajk )} are a set of t mutually orthogonal Latin squares of order q and C is the code with vectors {k, j, a1kj , · · · , atkj }, then there are no codes over Zq with the same parameters n, d as C having more than q 2 vectors. This follows from the fact that the minimum Hamming distance is t + 1 and the length is t + 2, so the Joshi bound forces the cardinality to be less than or equal to q 2 .

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Chinese Remainder Theorem

Let r and k be integers with k dividing r and define Ψk : (Z/rZ)n → (Z/kZ)n by Ψk (α1 , α2 , · · · , αn ) = (α1

(mod k), α2

(mod k), · · · , αn

(mod k))

where v = (α1 , α2 , · · · , αn ). Q For positive r = si=1 ki where gcd(ki , kj ) = 1 we define the map Ψ : (Z/rZ)n → (Z/k1 Z)n × (Z/k2 Z)n × · · · × (Z/ks Z)n by Ψ(v) := (Ψk1 (v), Ψk2 (v), · · · , Ψks (v)). Let C (k1 ) , C (k2 ) · · · C (ks ) be codes of length n, where C (ki ) is a code over Zki . We define the Chinese product by CRT(C (k1 ) , C (k2 ) , · · · , C (ks ) ) = {Ψ−1 (v1 , v2 , · · · , vs ) | vi ∈ C ki }, 5

where Ψ−1 (v1 , v2 , . . . , vs ) is the unique vector in (Z/Zr )n that is congruent componentwise to vi (mod ki ). The generalized Chinese Remainder Theorem gives that CRT is the inverse image of the map Ψ. See [4] for an application of this map to self-dual codes. Lemma 2.1 Let C (k1 ) , C (k2 ) , · · · , C (ks ) be codes over Zk1 , Zk2 , · · · , Zks respectively. Then rank(CRT(C (k1 ) , C (k2 ) , · · · , C (ks ) )) = Max{rank(C (ki ) )}. Proof. By the fundamental theorem of finitely generated abelian groups, C (k1 ) , C (k2 ) , · · · , C (ks ) are isomorphic to Zk1 /f11 Zk1 ⊕ Zk1 /f21 Zk1 ⊕ · · · ⊕Zk1 /fn1 Zk1 , Zk2 /f12 Zk2 ⊕ Zk2 /f22 Zk2 ⊕ · · · ⊕Zk2 /fn2 Zk2 , .. . Zks /f1s Zks ⊕ Zks /f2s Zks ⊕ · · · ⊕Zks /fns Zks , respectively, where f1i , f2i , · · · , fni are positive integers such that f1i |f2i | · · · |fni |ki for any i ∈ {1, 2, · · · , s}. Then by a property of elementary divisors, CRT(C (k1 ) , C (k2 ) , · · · , C (ks ) ) is also isomorphic to Zk1 k2 ···ks /(f11 f12 · · · f1s )Zk1 k2 ···ks ⊕ Zk1 k2 ···ks /(f21 f22 · · · f2s )Zk1 k2 ···ks ⊕ · · · · · · ⊕ Zk1 k2 ···ks /(fn1 fn2 · · · fns )Zk1 k2 ···ks . 2

This proves the lemma.

Theorem 2.2 Let C (k1 ) , C (k2 ) , · · · , C (ks ) be codes over Zk1 , Zk2 , . . . , Zks respectively. If C (ki ) is an MDR code for all i (not necessary the same rank), then CRT(C (k1 ) , C (k2 ) , · · · , C (ks ) ) is an MDR code. Proof : Let C := CRT(C (k1 ) , C (k2 ) , · · · , C (ks ) ). We have rank(C) = Max{rank(C (ki ) )}. So d(C) = min{d(C (ki ) )} = min{n − rank(C (ki ) ) + 1} = n − Max{rank(C (ki ) )} + 1 = n − rank(C) + 1. 2 (1) An example of the application of Theorem 2.2 Let C (4) and C (9) be the MDR codes over Z4 and Z9 with the generator matrices, 

G(4)



1 0 1 2    = 0 1 1 2   0 0 2 2 6

and





G(9)

1 0 1 2    =  0 3 0 3  0 0 3 3

respectively. Then the Chinese product code C of C (4) and C (9) has a generator matrix 



1 0 1 2    G =  0 21 9 30   0 0 30 30 and is an MDR code over Z36 . (2) A counterexample to the inverse of Theorem 2.2 Let C be an MDR code over Z6 with the generator matrix 



1 0 0 1    G=  0 3 0 3 . 0 0 3 3 Although Ψ2 (C) is also a binary MDR code, Ψ3 (C) = {(0, 0, 0, 0), (1, 0, 0, 1), (2, 0, 0, 2)} is not an MDR code over Z3 .

2.1

Polynomial Codes

We shall make the standard connection between codes in Znk and the rings Zk [x]/(xn − 1), namely the residue class of the polynomial c(x) = c0 + c1 x + c2 x2 + · · · + cn−1 xn−1 corresponds to the code word c = (c0 , c1 , c2 , · · · , cn−1 ). Moreover multiplication by x corresponds to a cyclic shift. So, we can define a cyclic code of length n over Zk as an ideal of Zk [x]/(xn − 1). For generalizations of some standard results on cyclic codes over Zpm applied to Zk see [8]. We note that from (1) any code over Zk is isomorphic to (Zk /el Zk )rl ⊕ (Zk /el−1 Zk )rl−1 ⊕ · · · ⊕ (Zk /e0 Zk )r0 , for integers 1 = e0 |e1 | · · · |el−1 |el = k. In this paper, we shall deal with a cyclic code C over Zk of length n with generators of the form {f0 (x)f2 (x) · · · fl (x); (k/el−1 )f0 (x)f1 (x)f3 (x) · · · fl (x); · · · ; (k/e1 )f0 (x)f1 (x) · · · fl−1 (x)}, 7

where f0 (x), f1 (x), · · · , fl (x) are pairwise-coprime polynomials such that f0 (x)f1 (x) · · · fl (x) = xn − 1. Then we note that |C| = k deg(f1 (x)) × (el−1 )deg(f2 (x)) × · · · × (e1 )deg(fl (x)) , rank(C) = deg(f1 (x)) + deg(f2 (x)) + · · · + deg(fl (x)) = n − deg(f0 (x)). In this case, we will say that the polynomial f0 (x) is the proper generator polynomial. Since cyclic codes are so important to the study of MDR codes we shall also show how the CRT applies to polynomial codes in their natural setting. Q Let k = ki where the ki are relatively prime. Define Φki : Zk [x]/(xn − 1) → Zki [x]/(xn − 1) such that Φki (a0 + a1 x + · · · + an−1 xn−1 ) = a0

(mod ki ) + a1

(mod ki )x + . . . + an−1

(mod ki )xn−1

Then we define Φk : (Zk [x]/(xn − 1)) → (Zk1 [x]/(xn − 1) × Zk2 [x]/(xn − 1) × · · · × Zks [x]/(xn − 1)) by Φk (p(x)) := (Φk1 (p(x)), Φk2 (p(x)), · · · , Φks (p(x))) Then for ideals Ij in Zkj [x]/(xn −1) we define the inverse image of the map by CRT(I1 , I2 , · · · , Is ), that is, CRT(I1 , I2 , · · · , Is ) is the unique ideal in Zk [x]/(xn − 1) that is congruent to Ij in Zkj [x]/(xn − 1). By the generalized Chinese Remainder Theorem, this map is well defined and furthermore CRT(I1 , I2 , · · · , Is ) is an ideal in Zk [x]/(xn − 1). Associating a cyclic code with its corresponding ideal we have the following proposition. Proposition 2.3 If {Ci } are cyclic codes over Zki [x]/(xn − 1) respectively, then CRT(C1 , · · · , Cs ) is a cyclic code over Zk [x]/(xn − 1). Proposition 2.4 Let p1 , p2 , · · · , ps be distinct primes and let g1 (x), g2 (x), · · · , gs (x) be fac−1 tors of xn − 1 in Zpm 1 [x], Zpm2 [x], · · · , Zpms [x] respectively. Then Φk (g1 (x), g2 (x), · · · , gs (x)) s 1 2 Q i is also a factor of xn − 1 in Zk [x], where k = si=1 pm i . n n n n Proof. Since Φ−1 k (x − 1, x − 1, · · · , x − 1) = x − 1, n n n Φ−1 k ((x − 1)/g1 (x), (x − 1)/g2 (x), · · · , (x − 1)/gs (x))

= (xn − 1)/Φ−1 k (g1 (x), g2 (x), · · · , gs (x)). n n n Since Φ−1 k ((x −1)/g1 (x), (x −1)/g2 (x), · · · , (x −1)/gs (x)) ∈ Zk [x], the proposition follows. 2 Moreover we prove the following lemma.

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Lemma 2.5 For any ai , 1 ≤ ai ≤ mi , there is a unit u ∈ Zk such that, as a1 a2 as a2 a1 Φ−1 k (p1 , p2 , · · · , ps ) = (p1 p2 · · · ps )u.

Proof. By the definition of Φk , ai as a2 a1 Φ−1 k (p1 , p2 , · · · , ps ) ≡ pi

i (mod pm i )

Therefore there exists xi ∈ Zk with as ms a1 m1 as a2 a1 Φ−1 k (p1 , p2 · · · , ps ) = p1 x1 + p1 = · · · = ps xs + ps

giving that s −as 1 −a1 + 1). + 1) = · · · = pas s (pm pa11 (pm s 1

2

i −ai Since each pm + 1 is relatively prime to each pi , the lemma follows. i

Let p1 , p2 , · · · , ps be distinct primes such that gcd(pi , n) = 1 for all pi and let C (p1 ) , C (p2 ) , · · · , C (ps ) be cyclic codes of length n over Zpm , Zpm , · · · , Zpm respectively. We take pairwises 1 2 (i) (i) (i) (i) (i) coprime polynomials f0 (x), f1 (x), · · · , fm (x) in each Zpm [x] such that f0 (x)f1 (x) · · · i (i) (i0 ) (i) (x) = xn − 1 and deg(fj (x)) = deg(fj (x)) for all j and 1 ≤ i, i0 ≤ s. Furthermore · · · fm (i) we define Fbj (x) ∈ Zpm [x] by i (i)

(i)

(i)

(i)

(i)

(i) Fbj (x) := f0 (x)f2 (x) · · · fj−1 (x)fj+1 (x) · · · fm (x).

Then we have the following theorem. Theorem 2.6 If each C (pi ) has generators of the form (i)

(i)

Fbm(i) (x)}, {Fb1 (x); pi Fb2 (x); · · · ; pm−1 i Qs

then CRT(C (p1 ) , C (p2 ) , · · · , C (ps ) ) is a cyclic code of length n over Zk , where k = ( with generators of the form (1)

(s)

(1)

i=1

pi )m ,

(s)

−1 b b b b {Gk } := {Φ−1 k (F1 (x), · · · , F1 (x)); (p1 · · · ps )Φk (F2 (x), · · · , F2 (x)); · · ·

b (1) b (s) · · · ; (p1 · · · ps )m−1 Φ−1 k (Fm (x), · · · , Fm (x))}.

Proof. Let C be the cyclic code of length n over Zk with generators of the form Gk , we denote C = h{Gk }i. Since Φ−1 k is a homomorphism and from the above lemma, C = h{Gk }i (1)

(s)

(1)

(s)

−1 b b b b = h{Φ−1 k (F1 (x), · · · , F1 (x)); Φk (p1 , · · · , ps )Φk (F2 (x), · · · , F2 (x)); · · ·

b (1) b (s) · · · ; Φk (pm−1 , · · · , pm−1 )Φ−1 1 s k (Fm (x), · · · , Fm (x))}i (1)

(s)

Fbm(s) (x))}i = h{Φ−1 (Fb1 (x), · · · , Fb1 (x)); · · · ; Φ−1 (pm−1 Fbm(1) (x), · · · , pm−1 s 1 0 0 0 := h{Fb1 (x); Fb2 (x); · · · ; Fbm (x)}i.

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If we take a polynomial c(x) ∈ C, that is, 0

0

0

c(x) = a1 (x)Fb1 (x) + a2 (x)Fb2 (x) + · · · + am (x)Fbm (x), then Φk (c(x)) (1)

(1)

Fbm(1) (x), · · · = (Φpm (a1 (x))Fb1 (x) + Φpm (a2 (x))p1 Fb2 (x) + · · · + Φpm (am (x))pm−1 1 1 1 1 (s)

(s)

(am (x))pm−1 Fbm(s) (x)) (a2 (x))ps Fb2 (x) + · · · + Φpm · · · , Φpm (a1 (x))Fb1 (x) + Φpm s s s s ∈ (C (p1 ) , C (p2 ) , · · · , C (ps ) ). Since Φk is isomorphism, c(x) ∈ CRT(C (p1 ) , C (p2 ) , · · · , C (ps ) ). Moreover we note that −1

|C| = (p1 · · · ps )m(n−deg(Φk = (p1 · · · ps ) Pm−1

= p1

j=0

Pm−1 j=0

(1)

(s)

−1

b1 (x),···,Fb1 (x))) (F (1)

× · · · × (p1 · · · ps )(n−deg(Φk

(1)

(s)

bm (x),···,Fbm (x))) (F

(s)

(m−j) deg(Φ−1 (fj+1 (x),···,fj+1 (x))) k

Pm−1

(1)

(m−j) deg(fj+1 (x))

× · · · × ps

j=0

(s)

(m−j) deg(fj+1 (x))

= |C (p1 ) | × · · · × |C (ps ) | = |CRT(C (p1 ) , C (p2 ) , · · · , C (ps ) )|. 2

3

BCH Codes and Reed Solomon Codes

In this section, we shall generalize the constructions of BCH codes and Reed-Solomon codes Q i to the ring Zk , k = si=1 pm i , where all pi are distinct primes. Let n be a divisor of φr (k) := Q k r si=1 (1 − 1/pri ) for a positive integer r, where φr denotes the generalized Euler function and let C be a cyclic code of length n over Zk with proper generator polynomial f0 (x). We define ms 1 Rr,k := GR(pm 1 , r) ⊕ · · · ⊕ GR(ps , r), ∗ i where GR(pm i , r) denotes the Galois ring of degree r (cf. p. 308 of [10]). Let Rr,k be the group of units of Rr,k . Since ∗ ms ∗ 1 R∗r,k = GR(pm 1 , r) × · · · × GR(ps , r) mi ∗ i (cf. p. 398 of [10]), where GR(pm i , r) is the group of units of GR(pi , r), we note that

|R∗r,k | = φr (k). 10

We shall take (if there exists) a unit α ∈ R∗r,k such that αn ≡ 1(modk), αj is not 1 for j < n, (which gives that αi 6= αj for i 6= j, i, j < n), 1 − αj is also a unit for j = 1, 2, · · · , n − 1, and that the least common multiple of the minimal polynomials of {αi , αi+1 , · · · , αi+δ−2 } is a factor of xn − 1, for some i and a positive integer δ(< n). We note that if n is a divisor of pr − 1, when k = pm , we can take α as a primitive nth root of unity in the Galois ring GR(pm , r). Definition 1 A cyclic code C of length n over Zk is a BCH code of designed distance δ if and only if the proper generator polynomial f0 (x) is, for some i, the least common multiple of the minimal polynomials of {αi , αi+1 , · · · , αi+δ−2 }. If k is a prime then we are in the classical case. Often we shall take i = 1. We know the following result as BCH bound (cf. [11]). Proposition 3.1 The minimum Hamming weight of a BCH code C of designed distance δ, is at least δ. Definition 2 The Reed-Solomon code (RS) over Zk is defined as a BCH code with r = 1 and i = 1. Then we have the following theorem. Theorem 3.2 A Reed-Solomon code over Zk is also an MDR code. Proof: Since α ∈ R1,k ms 1 = GR(pm 1 , 1) ⊕ · · · ⊕ GR(ps , 1)

= Zpm 1 ⊕ · · · ⊕ Zpms = Zk , s 1 the minimal polynomial of αj is simply x − αj , for j = 1, 2, · · · , δ − 1. So f0 (x) = (x − α)(x − α2 ) · · · (x − αδ−1 ). Thus rank(C) = n − deg(f0 (x)) = n − (δ − 1). Since d(C) ≤ n − rank(C) + 1 = δ, we have d(C) = δ and d(C) = n − rank(C) + 1. 2 We extend the RS code in standard way, namely if c = c0 , c1 , · · · , cn−1 then cn = −

n−1 X

ci

i=0

Theorem 3.3 An extended RS code is an MDR code. 11

Proof. To show that this produces an MDR code we only have to show that the minimum weight increases by 1. Any vector c corresponds to a polynomial of the form: c(x) = al (x)f0 (x) + al−1 (x)(k/el−1 )f0 (x) + · · · + a1 (x)(k/e1 )f0 (x) giving c(x) = (al (x) + al−1 (x)(k/el−1 ) + · · · + a1 (x)(k/e1 ))f0 (x) So c(x) = a(x)f0 (x) for some polynomial a(x). Let c(x) be the polynomial c0 + c1 x + c2 x2 + · · · + cn−1 xn−1 then c(1) = −cn . The standard proof applies, i.e. if c(x) = a(x)f0 (x) then f0 (1) is not 0. If a(1) 6= 0 then c(1) 6= 0 otherwise a(x)f0 (x) = b(x)(x − 1)f0 (x) and the BCH bound given in Proposition 3.1 applies. 2 We shall give an example for Z65 . Let Z65 := {0, 1, · · · , 64}. If n = 4, 8 is a primitive nth root of unity in Z65 . Since 82 ≡ 64 = −1 and 83 ≡ 57 = −8, we have x4 − 1 = (x − 8)(x − 82 )(x − 83 )(x − 1) = (x − 8)(x − 64)(x − 57)(x − 1) = (x − 8)(x + 1)(x + 8)(x − 1). Example 1: We take f0 (x) = (x − 8)(x + 1) = x2 − 7x − 8 f1 (x) = (x + 8)(x − 1) = x2 + 7x − 8, f2 (x) = 1. Let C be the code such that C = h{f0 (x)}i. We note that the code is free. Then rank(C) = 4 − deg(f0 (x)) = 2 and d(C) = 3 = 4 − 2 + 1. So the code is FMDR. Example 2: We take f0 (x) = (x − 8)(x + 1) = x2 − 7x − 8 f1 (x) = x + 7, f2 (x) = x − 1. Then f0 (x)f2 (x) = (x − 8)(x + 1)(x − 1) = x3 − 8x2 − x + 8, f0 (x)f1 (x) = (x − 8)(x + 1)(x + 8) = x3 + x2 + x + 1, Let C be the code such that C = h{f0 (x)f2 (x); 13f0 (x)f1 (x)}i. We note that the code is non-free. Then rank(C) = 4 − deg(f0 (x)) = 2 and d(C) = 3 = 4 − 2 + 1. So the code is 12

MDR. Example 3: We take f0 (x) = x − 8, f1 (x) = x + 1, f2 (x) = (x + 8)(x − 1). Then f0 (x)f2 (x) = (x − 8)(x + 8)(x − 1) = x3 + −1x2 + x − 1, f0 (x)f1 (x) = (x − 8)(x + 1) = x2 − 7x − 8, Let C be the code such that C = h{f0 (x)f2 (x); 13f0 (x)f1 (x)}i. We note that the code is also non-free. Then rank(C) = 4 − deg(f0 (x)) = 3 and d(C) = 2 = 4 − 3 + 1. So the code is also MDR.

3.1

Chinese Remainder Theorem of RS Codes

We shall show how the CRT construction applies to RS codes. Let n be a divisor of Qs mi ms 1 m gcd(φ1 (pm 1 ), · · · , φ1 (ps )), with k = i=1 pi , where αi is an element of Zpi i satisfying j the conditions given above. Namely, αi is a unit, αin = 1, αi 6= 1 for j < n, and 1 − αij is a unit for j = 1, 2, · · · , n − 1. Lemma 3.4 α = Φ−1 k (α1 , · · · , αs ) has the desired properties in Zk . Proof. It is clear that α is a unit. If αj were 1 for j < n then that would imply that αij −1 n n n was 1 in Zpm giving a contradiction. Moreover (Φ−1 k (α1 , · · · , αs )) = Φk (α1 , · · · , αs ) = 1 i since Φ−1 k is a ring isomorphism. Then it is clear that 1 − αj is a unit for j = 1, . . . , n − 1. 2 (pi ) Let {C } be the cyclic codes given in Theorem 2.6 respectively. Theorem 3.5 If {C (pi ) } are Reed-Solomon codes of designed distance δ, then CRT(C (p1 ) , · · · · · · , C (ps ) ) is also a Reed-Solomon code of designed distance δ. Proof. From Proposition 2.4 and Theorem 2.6, we can take a proper generator polynomial 0 f0 (x) of CRT(C (p1 ) , · · · , C (ps ) ) as 0

(1)

(s)

f0 (x) = Φ−1 k (f0 (x), · · · , f0 (x)). (i)

Since f0 (x) = (x − αi )(x − αi2 ) · · · (x − αiδ−1 ), for all i and by the above lemma, 0

−1 δ−1 δ−1 f0 (x) = Φ−1 k (x − α1 , · · · , x − αs ) · · · Φk (x − α1 , · · · , x − αs )

= (x − α)(x − α2 ) · · · (x − αδ−1 ). 13

2

The theorem follows.

Acknowledgment: The authors would like to thank the referee for a careful reading.

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