Maximum Distance Codes in M atn,s(Zk ) with a Non-Hamming Metric and Uniform Distributions Steven T. Dougherty Department of Mathematics University of Scranton Scranton, PA 18510 USA Keisuke Shiromoto ∗ Department of Mathematics Kumamoto University 2-39-1, Kurokami Kumamoto 860-8555 Japan June 22, 2011

∗

Research Fellow of the Japan Society for the Promotion of Science.

1

Abstract We establish a bound on the minimum ρ distance for codes in M atn,s (Zk ) with respect to their ranks and call codes meeting this bound MDR codes. We extend the relationship between codes in M atn,s (Zk ) and distributions in the unit cube and use the Chinese Remainder Theorem to construct codes and distributions.

Keywords: MDR codes, ρ metric, uniform distributions.

Contact author: Steven Dougherty Address: Department of Mathematics University of Scranton Scranton, PA 18510 Telephone: 570-941-6104 Fax: 570-941-5981 E-mail: [email protected]

2

1

Introduction

MDS codes in M atn,s (Fq ) with the ρ metric were defined by Rosenbloom and Tsfasman in [7] and related to possible information theory applications. In [9], Skriganov related these codes to uniform distributions and furthered their theory. In [2], Dougherty and Skriganov developed a weight enumerator for codes with the ρ metric and MacWilliams relations for this weight enumerator were given. In [3], they investigated these codes over an arbitrary alphabet and showed that over an arbitrary alphabet the weight spectra of an MDS code is fixed. Interestingly, these codes not only have a very strong relationship to uniform distributions but to association schemes as well; see [5] by Martin and Stinson for a description. On the other hand, Shiromoto and Yoshida [8] proved a bound on minimum Hamming weights for linear codes over Zk based on the ranks of the codes. In [1], Dougherty and Shiromoto studied codes over Zk meeting this bound, namely, MDR codes. In the present work we consider codes in M atn,s (Zk ) with the ρ metric and find a bound based on the rank of the code as a Zk -module. We extend the relation between codes and uniform distributions, and apply the Chinese Remainder Theorem to produce new codes and distributions. Throughout this paper, we shall mean that p is a prime, q is a prime power and k, n and s are positive integers.

1.1

Definitions and Notations

We begin with some definitions. Let M atn,s (Zk ) denote the linear space of all matrices with n rows and s columns with entries from the finite ring Zk := Z/kZ. A code is a subset of M atn,s (Zk ) and if it is a Zk -submodule then we say that the code is linear and, in particular, if it is a free Zk -submodule then we say the code is free. For two vectors v = (v1 , v2 , . . . , vs ), w = (w1 , w2 , . . . , ws ) in M at1,s (Zk ), define the map P ψ(v, w) = si=1 vi ws−i+1 . To the ambient space M atn,s (Zk ) we attach the following inner product: [Ω, Ω0 ] =

n X

ψ(Ωi , Ω0i ),

i=1

Ω0i

0

where Ωi and are the i-th row of Ω and Ω , respectively. The reason that the inner product is defined like this will become apparent after the connection to uniform distributions is described. Moreover, the MacWilliams relations given in [2] are for this inner product. For any code C ∈ M atn,s (Zk ) we define C ⊥ = {Ω | [Ω, Ω0 ] = 0 for all Ω0 ∈ C}. An equivalent metric to the ρ metric first appeared in [7] and was also defined in [9]. We shall now define this metric. For a non-zero vector ω = (ω1 , . . . , ωs ) ∈ M at1,s (Zk ) we have ρ(ω) = max{i | wi 6= 0}. If ω = 0 then ρ(0) = 0. Then for Ω ∈ M atn,s (Zk ), we define 3

ρ(Ω) = ni=1 ρ(Ωi ) where Ωi is the i-th row of Ω. For a linear code C in M atn,s (Zk ), the minimum ρ weight of C is defined by P

ρ(C) = min{ρ(Ω) | Ω ∈ C − {0}}, where 0 denotes the matrix in M atn,s (Zpm ) whose entries are all zero. Define the ρ weight enumerator by: X

P (C, z) =

z ρ(Ω) .

Ω∈C

MacWilliams relations for the ρ weight enumerator exist for codes in M at1,s (Fq ) and are given in [9]. However, it is shown in [2] that they do not extend. However, MacWilliams relations are given for a generalized version of this weight enumerator in [2] for codes in M atn,s (Fq ). Let C be a linear code in M atn,s (Zk ). Elements Ω1 , Ω2 , . . . , Ωt ∈ C are called generators P of C if C = ti=1 Zk Ωi . The rank of C, denoted by rank(C), is the minimum number of generators of C. The free rank of C, denoted by f-rank(C), is the maximum of the ranks of Zk -free submodules of C. We note that rank(C) + f-rank(C ⊥ ) = ns,

(1)

(see [8] and [1]). In particular, in M atn,s (Zpm ), there exist generators h

m−1 Ω10 , . . . , Ωh0 0 ; pΩ11 , . . . , pΩh1 1 ; . . . ; pm−1 Ω1m−1 , . . . , pm−1 Ωm−1

(2) of C such that

h0 + h1 + · · · + hm−1 = rank(C) h0 = f-rank(C). In this case, we note that |C| = (pm )h0 × (pm−1 )h1 × · · · × phm−1 . For a linear code C in M atn,s (Zk ), we define Cf as a free Zk -submodule of C with rank(Cf ) = f-rank(C) and define CF as a free Zk -submodule of M atn,s (Zk ) with rank(CF ) = rank(C) and C ⊆ CF . It follows that if C is a free code, then Cf = CF = C.

2 2.1

Maximum Distance Codes MDS Codes in M atn,s (Zk )

In this subsection we shall extend the definition of MDS codes to codes in M atn,s (Zk ) and in the next we shall establish a stronger bound and define MDR codes. We begin by stating a well known bound. 4

Proposition 2.1 ([9], [7]) If C is a linear code in M atn,s (Fq ) with dimension h, where 0 ≤ h ≤ ns, then (3)

ρ(Ω) ≤ ns − h + 1.

In [3] it is shown that this bound holds for codes consisting of q h elements in M atn,s (A) where A is an arbitrary alphabet with q elements. In these papers a code meeting this bound is called Maximum Distance Separable. It is natural to name these codes MDS because of the similarity to the MDS bound for codes with the Hamming metric. In fact the standard bound is implied by this bound. We have the following, which follows from [3]. Theorem 2.2 If C is a code in M atn,s (Zk ) with k λ elements, where 0 ≤ λ ≤ ns, then ρ(C) ≤ ns − λ + 1. We say that a code in M atn,s (Zk ) is Maximum Distance Separable (MDS) if it meets this bound and call this bound the MDS bound. See [7], [9], [2], [3] for a description of these codes.

2.2

MDR Codes in M atn,s (Zpm )

For codes over rings, we can also consider the rank of the code. Namely, we are interested in how this bound is affected by the rank of the code and not simply by the cardinality of the code. For codes over fields the dimension is uniquely determined by the cardinality, but for codes over rings the situation is more complex. In general, bounds on the minimum weight are effected by the rank and the cardinality. For a linear code C in M atn,s (Zpm ), we define S(C) := {Ω ∈ C | pΩ = 0}. We note that if C is a free code in M atn,s (Zpm ), then S(C) = pm−1 C := {pm−1 Ω | Ω ∈ C}. Since Zpm is a local commutative ring (cf. [6]), we have that if C is a linear code with rank h in M atn,s (Zpm ), then S(C) can be viewed as a linear code with dimension h in M atn,s (Fp ). We have the following lemma. Lemma 2.3 Let C be a linear code in M atn,s (Zpm ). Then ρ(C) = ρ(S(C)) = ρ(CF ).

5

Proof. We have that S(C) ⊆ C and so ρ(S(C)) ≥ ρ(C). If even one minimum weight vector of C is in S(C) then ρ(C) = ρ(S(C)) since this vector would have ρ weight equal to ρ(C). We shall show that at least one minimum weight vector of C must be in S(C). Assume otherwise and let Ω ∈ C such that ρ(C) = ρ(Ω) and Ω 6∈ S(C). It follows that there exists i, 1 ≤ i ≤ m − 1, such that pi Ω 6= 0 and pi+1 Ω = 0. We set Ω0 := pi Ω. Since Ω0 ∈ S(C) and we have assumed that ρ(S(C)) > ρ(C) then ρ(Ω0 ) > ρ(Ω). This is a contradiction since ρ(Ω) ≥ ρ(pi Ω). Hence ρ(C) = ρ(S(C)). On the other hand, since S(C) ⊆ S(CF ) and |S(C)| = |S(CF )| = ph , where h is the rank of the code, we have S(C) = S(CF ). The lemma follows. 2 This leads naturally to the following theorem. Theorem 2.4 If C is a linear code with rank h in M atn,s (Zpm ) then ρ(C) ≤ ns − h + 1. Proof. Using Lemma 2.3, the theorem follows from bound (3) for S(C).

2

We shall call codes meeting this bound as Maximum Distance with respect to Rank (MDR) Codes and call this bound the MDR bound. See [1] for similar results on codes in Znk with respect to the Hamming metric. We shall give an example of an MDR code in the following example. Example 1. Let C be the linear code in M at2,2 (Z4 ) generated by 1 0 0 1

!

and

0 2 2 2

!

.

We have that rank(C) = 2 and ρ(C) = 3, so C is an MDR code in M at2,2 (Z4 ).

2.3

Weight Spectra of MDR Codes

It is well known that the weight enumerator of an MDS code with respect to the Hamming weight has a uniquely determined weight enumerator. The case is similar for an MDS code with respect to the ρ metric. We shall define the weight spectra of a code and state a known result showing how it is determined for MDS codes. For a linear code C in M atn,s (Zpm ), the i-th weight spectra of C is defined as follows: AC (i) = |{Ω ∈ C | ρ(Ω) = i}|, where 0 ≤ i ≤ ns. In [3], the i-th weight spectra of a code consisting of q h elements in M atn,s (A), where A is an arbitrary alphabet, is determined as follows: 6

Proposition 2.5 Let C be an MDS code with q h elements in M atn,s (A). Then AC (0) = 1, AC (i) = 0 for 1 ≤ i < ρ(C) = ns − h + 1, and AC (i) =

n X l−1

i−ρ(C) X l n (q i−ρ(C)+1−t − 1), σs (l, i) (−1)t t l t=0

!

!

for ρ(C) ≤ i ≤ ns, where σs (l, i) := |{(a1 , a2 , . . . , al ) ∈ Nl |

l X

aj = i, 0 < aj ≤ s, 1 ≤ j ≤ l}|.

j=1

Of course, linear codes need not have q h elements. Thus, we are interested in the weight spectra of a linear MDR code. For MDR codes in M atn,s (Zpm ), we have the following bounds on weight spectras. Theorem 2.6 Let C be a linear code with rank h in M atn,s (Zpm ). If C is an MDR code, then AC (0) = 1, AC (i) = 0 for 1 ≤ i < ρ(C) = ns − h + 1, and n X l=1

i−ρ(C) X l n (−1)t (pi−ρ(C)+1−t − 1) ≤ AC (i) ≤ σs (l, i) t l t=0

!

!

n X l=1

i−ρ(C) X n l σs (l, i) (−1)t (pm(i−ρ(C)+1−t) − 1), l t t=0

!

!

for ρ(C) ≤ i ≤ ns. Proof. Since S(C) ⊆ C ⊆ CF , we have AS(C) (i) ≤ AC (i) ≤ ACF (i) for all i. The theorem follows from Proposition 2.5. 2 This gives the following corollary. Corollary 2.7 If C is an MDR code in M atn,s (Zpm ), then  

n σs (l, ρ(C) + 1)l l Pn n l=1 l σs (l, ρ(C) + 1)

Pn

l=1

≤ p + 1.

Proof. We put i = ρ(C) + 1 in AS(C) (i). Then AS(C) (i) =

n X l=1

!

n σs (l, ρ(C) + 1)(p − 1)(p + 1 − l) ≥ 0. l 2

So we have the above inequality.

7

2.4

Duality for MDR Codes

For MDS codes in the Hamming and ρ metrics it is known that the orthogonals of MDS codes are MDS codes. In this subsection we shall give corresponding relationships for MDR codes. We begin with some definitions. We define the map φ from the ring pm−1 Zpm := pm−1 (Z/pm Z) = {0, pm−1 , 2pm−1 , . . . , (p− 1)pm−1 } to the ring (field) Zp = {0, 1, · · · , p − 1} as follows: φ : pm−1 Zpm −→ Zp apm−1 7−→ a. We note that for x, y ∈ Zpm , φ(pm−1 (x + y)) = φ(pm−1 x) + φ(pm−1 y) and φ(pm−1 (xy)) = φ(pm−1 x)φ(pm−1 y) in Zp and the map gives an isomorphism. Moreover, we extend the map to the matrix space. We define the map φn,s as follows: φn,s : M atn,s (pm−1 Zpm ) −→ M atn,s (Zp ) (Ωi,j ) 7−→ (φ(Ωi,j )), where (Ωi,j ) denotes the matrix which has the (i, j)-th entry Ωi,j . We note that if C is a linear code in M atn,s (Zpm ), then φn,s (S(C)) is a linear code in M atn,s (Zp ). Lemma 2.8 For v, w ∈ M at1,s (Zpm ), φ(pm−1 ψ(v, w)) = ψ(φ1,s (pm−1 v), φ1,s (pm−1 w)) in Zp . Moreover, for Ω, Ω0 ∈ M atn,s (Zpm ), φ(pm−1 [Ω, Ω0 ]) = [φn,s (pm−1 Ω), φn,s (pm−1 Ω0 )] in Zp . Proof. The lemma follows in a straightforward manner from the properties of φ, φn,s and the inner product. 2 Now we have the following proposition. Proposition 2.9 For a linear code C with rank h in M atn,s (Zpm ), φn,s (S(C))⊥ = φn,s (S((C ⊥ )f )). Proof. Suppose that Ω ∈ (C ⊥ )f . Since pm−1 Ω ∈ S((C ⊥ )f ) and [Ω, Ω0 ] = 0 in Zpm for all Ω0 ∈ S(C) (⊆ C), we have [φn,s (pm−1 Ω), φn,s (Ω0 )] = 0

8

in Zp for all φn,s (Ω0 ) ∈ φn,s (S(C)) by Lemma 2.8. Thus φn,s (S((C ⊥ )f )) ⊆ φn,s (S(C))⊥ . Since |φn,s (S((C ⊥ )f ))| = |φn,s (S(C))⊥ | = pns−h 2

then the proposition follows from equation (1). In [9], the duality for MDS codes in M atn,s (Fq ) is given as follows: Lemma 2.10 If C is a linear MDS code in M atn,s (Fq ), then C ⊥ is also an MDS code. Using the above results, we have a duality for MDR codes in M atn,s (Zpm ).

Theorem 2.11 Let C be a linear code in M atn,s (Zpm ). If C is an MDR code, then (C ⊥ )f is also an MDR code. Proof. Suppose that C is an MDR code. Then φn,s (S(C)) and φn,s (S(C))⊥ are MDR (MDS) codes in M atn,s (Zp ) by Lemma 2.10. Moreover, φn,s (S((C ⊥ )f )) is also an MDR code in M atn,s (Zp ) by Proposition 2.9. 2 Example 2. Let C be the code in M at2,2 (Z4 ) defined in Example 1. Then C ⊥ is the code generated by 1 3 1 0

!

,

0 3 1 1

!

and

2 0 0 0

!

.

Since rank(C ⊥ ) = 3 and ρ(C ⊥ ) = 1, we have that C ⊥ is not an MDR code. On the other hand, we can take a code (C ⊥ )f as the code generated by 1 3 1 0

!

and

0 3 1 1

!

.

Then (C ⊥ )f is an MDR code, since rank((C ⊥ )f ) = 2 and ρ((C ⊥ )) = 3. Given the MDR bound we shall determine when codes meeting this bound exist. First we state the following lemma which was proved in [9]. Lemma 2.12 For each 1 ≤ k ≤ ns, there exists an MDS code in M atn,s (Fq ) with q k elements, whenever n ≤ q + 1. Using this lemma, we have the following existence result for MDR codes in M atn,s (Zpm ). Theorem 2.13 There exist MDR codes in M atn,s (Zpm ) with rank h(1 ≤ h ≤ ns), whenever n ≤ p + 1. Proof. If n ≤ p + 1, then there exists an MDS code D in M atn,s (Zp ) from Lemma 2.12. Thus there exists a code C in M atn,s (Zpm ) such that S(C) = φ−1 n,s (D). The theorem follows. 2

9

2.5

Chinese Remainder Theorem and MDR codes

The Chinese Remainder Theorem is a very valuable tool in studying codes over rings. It has been used to create MDS codes and self-dual codes from existing codes. This theorem was first used to construct codes in M atn,s (Zk ) in [3]. Here we shall use it to construct MDR codes and then extend it to Uniform Distributions and relate the two maps via a canonical bridge. Let Θk0 : M atn,s (Zk ) → M atn,s (Zk0 ) where k 0 divides k, be defined as Θk0 (Ωij ) = Ωij

(mod k 0 ).

Let Θ : M atn,s (Zk ) → M atn,s (Zk1 ) × M atn,s (Zk2 ) × . . . × M atn,s (Zkr ) where k =

Qr

i=1

ki and gcd(ki , kj ) = 1 for i 6= j, be defined as Θ(Ω) = (Θk1 (Ω), Θk2 (Ω), . . . , Θkr (Ω)).

Denote the inverse of Θ by CRT, and note that by the generalized Chinese Remainder Theorem, CRT is a ring isomorphism. Since C = CRT(C1 , . . . , Cr ), where each Ci is a linear code in M atn,s (Zki ) is a Zk -module, it is well known that rank(C) = max{rank(Ci )} in this Q situation (cf. [1]). It follows that every code C in M atn,s (Zk ) with k = ri=1 pai i , where p1 , . . . , pr are distinct primes, is an image of CRT, i.e., C = CRT(C1 , C2 , . . . , Cr ) for some Ci in M atn,s (Zpai i ). Lemma 2.14 Let C1 , C2 , . . . , Cr be linear codes in M atn,s (Zk1 ), . . . , M atn,s (Zkr ), respectively, where k1 , . . . , kr are positive integers with gcd(ki , kj ) = 1 for i 6= j. Then ρ(CRT(C1 , C2 , . . . , Cr )) = min{ρ(Ci )}. Proof. If Ωi is an element in each Ci , then ρ(CRT(Ω1 , . . . , Ωr )) ≥ ρ(Ωi ) since if the element in CRT(C1 , C2 , . . . , Cr ) had ρ weight less than this, then there would exists an element Ωi in Ci for some i with ρ weight less than min{ρ(Ci )}. 2 We shall now extend the MDR bound to codes over any ring Zk . Let k = prime, pi 6= pj when i is not equal to j. Theorem 2.15 If C is a linear code in M atn,s (Zk ) of rank h, then ρ(C) ≤ ns − h + 1.

10

Q ai

pi with pi

Proof. Let C = CRT(C1 , C2 , . . . , Cr ) for each linear code Ci in M atn,s (Zpai i ). Assume ρ(C) > ns − h + 1. Then for some i, rank(Ci ) = h and we know that ρ(Ci ) ≥ ρ(C). Then we have that ρ(Ci ) ≥ ρ(C) > ns − rank(Ci ) + 1 which is a contradiction for the MDR bound. 2 Given this bound we can consider MDR codes in M atn,s (Zk ) as well as in M atn,s (Zpm ). In the next theorem we shall show how to construct MDR codes from existing MDR codes over rings. Theorem 2.16 Let C1 , C2 , . . . , Cr be linear codes in M atn,s (Zk1 ), . . . , M atn,s (Zkr ), respectively, where k1 , . . . , kr are positive integers with gcd(ki , kj ) = 1 for i 6= j. If Ci is an MDR code for all i, then C = CRT(C1 , C2 , . . . , Cr ) is an MDR code. Proof. We put C := CRT(C1 , C2 , . . . , Cr ). Then ρ(C) = min{ρ(Ci )} = min{ns − rank(Ci ) + 1} = ns − max{rank(Ci )} + 1 = ns − rank(C) + 1. 2

The theorem follows.

In [3], they give an existence result for MDS codes in M atn,s (Zk ). Similarly, we have an existence result for MDR codes in M atn,s (Zk ). Theorem 2.17 There exist MDR codes in M atn,s (Zk ) with rank h (1 ≤ h ≤ ns) where Q k = ri=1 piai and p1 , . . . , pr are distinct primes, if n ≤ pi + 1 for all pi . Proof. The theorem follows from the above discussion and Theorem 2.13.

2

We shall now examine the duality relation with respect to the Chinese Remainder Theorem. Specifically, we shall show how the orthogonal is obtained via the CRT and get a duality result similar to Theorem 2.11. Theorem 2.18 Let C1 , C2 , . . . , Cr be linear codes in M atn,s (Zk1 ), . . . , M atn,s (Zkr ), respectively, where gcd(ki , kj ) = 1 for i 6= j, and set C = CRT(C1 , C2 , . . . , Cr ). Then C ⊥ = CRT(C1⊥ , C2⊥ , . . . , Cr⊥ ). 

Proof. Let Ω =

     

Ω1 Ω2 .. .

      

 0

∈ C and Ω =

     

Ω01 Ω02 .. .

      

∈ C ⊥ , where each Ωi and Ω0i denote the i-th

Ωn Ω0n P row of Ω and Ω , respectively. Then we have that [Ω, Ω0 ] = 0 implies that ni=1 ψ(Ωi , Ω0i ) = 0. Then, since CRT is a ring isomorphism, 0

ψ(Ωi

(mod ki ), Ω0i 11

(mod ki )) = 0,

where Ωi (mod ki ) reads each coordinate mod ki . Then Ω0 ∈ CRT(C1⊥ , C2⊥ , . . . , Cr⊥ ) and C ⊥ ⊆ CRT(C1⊥ , C2⊥ , . . . , Cr⊥ ). Moreover, the cardinalities of the two sets are the same giving equality. 2 We can now give a duality result for MDR codes in M atn,s (Zk ). Theorem 2.19 Let C1 , C2 , . . . , Cr be linear codes in M atn,s (Zpa1 1 ), . . . , M atn,s (Zpar r ), respectively, where p1 , . . . , pr are distinct primes and set C = CRT(C1 , C2 , . . . , Cr ). If Ci is an MDR code for all i, then CRT((C1⊥ )f , . . . , (Cr⊥ )f ) (⊆ C ⊥ ) is also an MDR code. Proof. We put D := CRT((C1⊥ )f , . . . , (Cr⊥ )f ). Since ρ(D) = min{ρ((Ci⊥ )f )} and (Ci⊥ )f is an MDR code for all i, we have ρ(D) = min{ns − rank((Ci⊥ )f ) + 1} = min{rank(Ci )} + 1, by using the equation (1). On the other hand, we have ns − rank(D) + 1 = ns − max{rank((Ci⊥ )f )} + 1 = ns − max{ns − rank(Ci )} + 1 = min{rank(Ci )} + 1. 2

The theorem follows.

Remark. In the above theorem, if rank(C) = rank(Ci ) for all i, then we can take a free code (C ⊥ )f as CRT((C1⊥ )f , . . . , (Cr⊥ )f ).

2.6

Relative Strength of the Two Bounds

In this subsection we shall compare the MDR and the MDS bounds. Lemma 2.20 For a non-free linear code C in M atn,s (Zpm ), the MDR bound is stronger than the MDS bound. These two bounds coincide for a free code. Proof. If C is a linear code in M atn,s (Zpm ) as in (2), then rank(C) = h0 + h1 + · · · + hm−1 1 1 ≥ h0 + h1 + · · · + m−1 hm−1 p p = logpm |C|. 2

The lemma follows.

12

Theorem 2.21 For a non-free linear code C in M atn,s (Zk ), the MDR bound is stronger than the MDS bound. These two bounds coincide for a free code. Proof. Put C := CRT(C1 , C2 , . . . , Cr ), where each Ci is a linear code in M atn,s (Zpai i ) of Q Q rank hi and k = ri=1 pai i with pi prime, pi 6= pj when i is not equal to j. Since |C| = ri=1 |Ci | and from Lemma 2.20 r |C| ≤

Y

(pai i )hi ≤ k h ,

i=1

2

where h = max{hi } = rank(C). The theorem follows.

Corollary 2.22 All MDS codes in M atn,s (Zk ) are free codes. Proof. From the above theorem, if C is an MDS code, then the equality in the proof of the previous theorem holds, that is, rank(Ci ) = f-rank(Ci ). 2 As an example, consider a code of rank 2 in M at2,2 (Z4 ) then ρ(C) ≤ 3, by the MDR bound. Let C be the code C={

0 0 0 0

!

,

2 0 0 2

!

0 2 2 0

!

2 2 2 2

!

}.

This code is an MDR code, but does not meet the bound. Specifically, 4 − log4 4 + 1 = 4 and therefore the code is not MDS. Moreover, the MDR bound can be applied naturally to all linear codes. In contrast, the MDS bound applies only to codes with |Zk |λ elements. For example, the code C in M at2,2 (Z8 ) C={

0 0 0 0

!

,

0 4 0 4

!

}

has 2 elements and rank 1, with ρ(C) = 4 and thus is an MDR code. However, the code does not have 8λ elements for some integer λ and the MDS bound does not directly apply.

3

Uniform Distributions

In this section we shall define uniform distributions and show how they relate to the codes we have constructed. In [9], a bridge between M atn,s (Fq ) and Uniform Distributions is made. We shall now extend that bridge from M atn,s (Zk ) to a wider class of uniform distributions. We begin with some definitions. mn mn +1 m1 m1 +1 n Let U denote the interval [0, 1) and ∆M A = [ ka1 , ka1 ) . . . [ kan , kan ) ⊂ U an elementary box, where M = (m1 , . . . , mn ) and A = (a1 , . . . , an ). 13

Definition 1 Given integers 0 ≤ δ ≤ s, a subset D ⊂ U n consisting of k δ−s points is called δ−s contains a (δ, s, n)-net of deficiency δ in base k if each elementary box ∆M A of volume k δ exactly k points of D. Definition 2 Given an integer 0 ≤ h ≤ n, a subset D ⊂ U n consisting of k h points is called −h an optimum [ns, h]s distribution in base k if each elementary box ∆M contains A of volume k exactly one point of D. We shall show how the Chinese Remainder Theorem can be used in this setting in a Q manner similar to the one shown for codes. Let k = ki where the ki are pairwise relatively prime. Let x1 , x2 , . . . , xr be points in U n , where xi =

X j

ai ki −j

j

is the ki -ary representation of the point xi . As for codes, we define Θ0ki (x) = Θ(

X

αj k −j ) =

X

(αj

(mod ki ))ki−j

j

j

and define Θ0 (x) = (Θ0k1 (x), Θ0k2 (x), . . . , Θ0kγ (x)) Define by CRT0 (x1 , x2 , . . . , xr ) the inverse of Θ0 , that is CRT0 (x1 , x2 , . . . , xr ) is a point x, where X x= αj k −j j

and αj = crt(aj1 , aj2 , . . . , ajr ), where crt(aj1 , aj2 , . . . , ajr ) is the unique element mod k that is αij (mod ki ). The map CRT0 can also be easily restricted to points in Qn (kis ), where Q(kis ) is the set P of all points x in U , such that if x = ai (ki )−i then ai = 0 if i > s and Qn (kis ) is the n-fold product of these sets. Let D1 , D2 , . . . , Dr be distributions of points in U n . Define CRT0 (D1 , D2 , . . . , Dr ) = {CRT0 (x1 , x2 , . . . , xr ) | (x1 , x2 , . . . , xr ) ∈ D1 × D2 × · · · × Dr }. It is clear that 0

|CRT (D1 , D2 , . . . , Dr )| =

r Y i=1

14

|Di |.

Lemma 3.1 Let Ei be elementary intervals in base ki . i.e. s Y

Ei =

"

j=1

bij ki dj

,

bij + 1

!

ki dj

of volume ki T −M , let E be the elementary interval in base k s  Y aj

E=

j=1

k

, dj

aj + 1 k dj



such that CRT0 (b1j , b2j , . . . , brj ) = aj . Then the volume of E is k T −M . Moreover, (x1 , x2 , . . . , xr ) ∈ E1 × E2 × · · · × Er if and only if CRT0 (x1 , x2 , . . . , xr ) ∈ E. Proof. The volume of E follows from the fact that vol(Ei ) = kiT −M =

s Y

1

j=1

ki j

d

s  Y

=

j=1

1 ki

dj

1 = ki 



P

dj

for each i hence dj = T − M . This gives that vol(E) = k T −M . Let x = CRT0 (x1 , x2 , . . . , xr ) where x, x1 , x2 , . . . , xr ∈ U n and xi = (y1i , . . . , yni ) with yji ∈ U . We have that P

(x1 , . . . , xr ) ∈ E1 × E2 × · · · × Er ⇐⇒ xi ∈ Ei for all i ⇐⇒ ! i + 1 b j j yji ∈ dj , dj for all i and j ki ki " i b

Set yji = then yji ∈ [ yji ∈ [

bij d ki j

follows.

,

bij d ki j

bij +1 d

ki j

,

bij +1 d

ki j

)

⇐⇒

X i,j c k −h h

i

i,j i ci,j h = 0 for all h < dj and ch = bj for h = dj . Hence

2,j r,j ) ⇐⇒ crt(c1,j h , ch , . . . , ch ) is 0 if h < dj and aj for h = dj and the result

2

Theorem 3.2 Let D1 , D2 , . . . , Dr be distributions in U n of base ki with k = ki and 0 gcd(ki , kj ) = 1 if i 6= k. If Di is a (T, M, S)-net for each i then CRT (D1 , D2 , . . . , Dr ) is a (T, M, S)-net in base k. If Di is an optimal distribution for each i then CRT0 (D1 , D2 , . . . , Dr ) is an optimal distribution in base k. Q

15

Proof. Given any elementary interval E in base k, E=

s  Y aj j=1

Let Ei be Ei =

s Y j=1

"

aj

k

, dj

aj + 1 . k dj 

(mod ki ) (aj + 1) (mod ki ) , ki dj ki dj

!

and then the previous lemma applies. Let Yi be the set of points in Di that are in the elementary interval Ei and let Y = CRT0 (Y1 , Y2 , . . . , Yr ). It is clear that Y ⊂ E. Moreover, since |Yi | = kiT we have that |Y | = k T and we have the result. To prove the second statement take E and Ei as given above. Assume that the volume of E is k −h making the volume of Ei equal to (ki )−h . Assume Di are optimal distributions. If xi is the unique point of Di in Ei then CRT0 (x1 , . . . , xn ) ∈ E ⇐⇒ xi ∈ Ei for all i by the previous lemma. Hence there exists a unique point of D in E. 2 Note that in (Proposition 2.1 in [9]), Skriganov shows that an optimum [ns, h]s distribution with s ≤ h ≤ ns is a (h − s, h, n) net in the same base k. Hence using this with the first part of the theorem gives an alternate proof of the second part.

4

Connections

We shall describe how the two spaces relate, following the notation given in [9]. For a point X in Qn (k s ), define the following matrix which is an element of M atn,s (Zk ): ΩhXi = (ω(x1 ), ω(x2 ), . . . , ω(xn ))T where ωhxi = (ξ1 (x), ξ2 (x), . . . , ξs (x)) and x = si=1 ξi (x)k i−s−1 . In [9], this definition is given only when the underlying alphabet is a finite field, but the definition naturally generalizes to the one given above. Moreover it is clear that Ω−1 is well defined on the space Qn (Zsk ). Let P

Γ : M atn,s (Zk1 ) × M atn,s (Zk2 ) × · · · × M atn,s (Zkr ) → Qn (Zsk1 ) × Qn (Zsk2 ) × · · · × Qn (Zskr ) so that Γ(X1 , X2 , . . . , Xr ) = (Ω(X1 ), Ω(X2 ), . . . , Ω(Xr )). It is clear then that the following diagram commutes. 16

Qn (k1 s ) × Qn (k2 s) × · · · × Qn (kr s )  Γy

−−−−−→ CRT0

s Qn (R  )  Ωy

M atn,s (Zk1 ) × M atn,s (Zk2 ) × · · · × M atn,s (Zkr ) −−−−−→ M atn,s (Zk ) CRT In Theorem 2.1 of [9] the following is shown: If D is a distribution in Qn (q s ) where q is a prime power and C is its corresponding code, i.e. C = Ω(D), then the following two statements are equivalent: (i) D is an optimum [ns, k]s distribution (ii) C is an MDS code in the ρ metric. Note that this is proven only when q is a prime power. However, using that the diagram commutes and this theorem we have the following: Theorem 4.1 Let C be an optimum distribution in Qn (k s ) for any k and C its corresponding code then the following are equivalent: (i) D is an optimum [ns, λ]s distribution in base k (ii) C is an MDS code in the ρ metric in M atn,s (Zk ). Hence we have established a wider class of uniform distributions which we consider, namely uniform distributions in any base. Q In [3] it is shown that there exists MDS codes in M atn,s (Zk ) for k ≥ 2 with k = pai i , and pi ≥ n − 1, where the pi are distinct primes, with k λ elements for all h with 1 ≤ λ ≤ ns. This gives the following: Corollary 4.2 There exist optimum [ns, λ] distributions for base k, with k = pi ≥ n − 1, where the pi are distinct primes, for all h with 1 ≤ λ ≤ ns.

Q ai

pi , and

References [1] S.T. Dougherty and K. Shiromoto, MDR codes over Zk , IEEE-IT, Vol. 46, No. 1, (2000), pp. 265–269. [2] S.T. Dougherty and M.M. Skriganov, MacWilliams Duality and the RosenbloomTsfasman Metric, Moscow Math. Journal, Vol. 2, No.1, (2002), pp. 81-97. [3] S.T. Dougherty and M.M. Skriganov, Maximum Distance Separable Codes in a NonHamming Metric over Arbitrary Alphabets , to appear in the Journal of Algebraic Combinatorics. [4] F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes, NorthHolland, Amsterdam, (1977). 17

[5] W.J. Martin and D.R. Stinson, Association schemes for ordered orthogonal arrays and (T, M, S)-nets, Canad. J. Math. Vol. 51, No. 2, (1999), pp. 326-346. [6] B. R. McDonald, Finite Rings with Identity, New York: Marcel Dekker, (1974). [7] M. Yu Rosenbloom and M. A. Tsfasman, Codes for the m-metric, Problems of Information Transmission, Vol. 33, No. 1, (1997), pp. 45–52. (Translated from Problemy Peredachi Informatsii, Vol. 33, No. 1, (1996), pp. 55-63.) [8] K. Shirmoto and T. Yoshida, A Singleton bound for linear codes over Zl , (preprint). [9] M.M. Skriganov, Coding Theory and Uniform Distributions, Algebra i Analiz, Vol. 13, No. 2, (2001), pp. 191-239, (Russian). (English translation to appear in St. Petersburg Math. Journal.)

18