MacWilliams Duality and the Rosenbloom-Tsfasman Metric Steven T. Dougherty ∗ Department of Mathematics University of Scranton Scranton, PA 18510 USA Email: [email protected]

Maxim M. Skriganov † Steklov Institute of Mathematics at St. Petersburg Fontanka 27, St. Petersburg 191011 Russia Email: [email protected]

Abstract A new non-Hamming metric on linear spaces over finite fields has recently been introduced by Rosenbloom and Tsfasman [9]. We consider orbits of linear groups preserving the metric and show that weight enumerators suitably associated with such orbits satisfy MacWilliams-type identities for mutually dual codes. Furthermore, we show that the corresponding weight spectra of dual codes are related by transformations which involve multi-dimensional generalizations of known Krawtchouk polynomials. The relationships with recent results by Godsil [6] and Martin and Stinson [8] on MacWilliams-type theorems for association schemes and ordered orthogonal arrays are also briefly discussed in the paper. ∗

The author thanks the Euler and Steklov Institutes in St. Petersburg, Russia, where he stayed while most of this work was done. † The author was partially supported by the Russian fund for Fundamental Research (Project No. 99-0100106) and by INTAS (Grant No. 00 - 429).

1

1

Introduction

Let M atn,s (Fq ) denote the linear space of all matrices with n rows and s columns with entries from a finite field Fq of q elements. A linear code is a subspace of M atn,s (Fq ). By definition (cf. MacWilliams and Sloane [7]) the Hamming weight κ(Ω), Ω ∈ M atn,s (Fq ), is equal to the number of non-zero entries of a matrix Ω. In this case κ(Ω1 − Ω2 ) defines the Hamming metric on M atn,s (Fq ). Introduce the following non-Hamming weight ρ on M atn,s (Fq ). At first, let n = 1 and ω = (ξ1 , ξ2 , . . . , ξs ) ∈ M at1,s (Fq ). Then, we put ρ(0) = 0 and (1)

ρ(ω) = max{i | ξi 6= 0}

for ω 6= 0. Now let Ω = (ω1 , . . . , ωn )T ∈ M atn,s (Fq ), ωj ∈ M at1,s (Fq ), 1 ≤ j ≤ n, and (·)T denotes the transpose of a matrix. Then, we put (2)

ρ(Ω) =

n X

ρ(ωj )

j=1

One can easily check that ρ is a metric on M atn,s (Fq ). It is obvious that κ(Ω) ≤ ρ(Ω) ≤ sκ(Ω.) Moreover, these inequalities cannot be improved on the whole space M atn,s (Fq ). Thus, for s > 1 the metric ρ is stronger that the Hamming metric κ. For s = 1 the metrics ρ and κ coincide. In coding theory the metric ρ was first introduced by Rosenbloom and Tsfasman [9]. It is worth noting that in the context of the theory of uniform distributions the metric ρ was also introduced in papers [8] by Martin and Stinson and [10, 11] by Skriganov. As an application of the ρ metric we should mention a recent paper [2] by Chen and Skriganov where explicit constructions of point sets with minimal order of the L2 discrepancy are given as codes with large weights simultaneously in the Hamming and Rosenbloom-Tsfasman metrics. Previously the main results of this paper were given in our preprint [5]. We refer to [11] for further comments. It is remarkable that fundamental concepts related to the Hamming metric can be very naturally extended to the metric ρ. For example, the metric ρ enables a rich theory of maximum distance separable codes (see [11]). In the present paper we shall give the corresponding extensions for the classical MacWilliams identities. Conceivably, in view of many applications to different areas of combinatorial mathematics (cf. the papers cited above), the metric ρ could take great significance comparable to the Hamming metric. For a given linear code C ⊂ M atn,s (Fq ) the following set of nonnegative integers (3)

wr (C) = |{Ω ∈ C | ρ(Ω) = r}|, 0 ≤ r ≤ ns 2

is called the ρ weight spectrum of the code C, where |{·}| denotes the cardinality of the set. Define the ρ weight enumerator by W (C|z) =

(4)

ns X

wr (C)z r =

r=0

X

z ρ(Ω)

Ω∈C

Introduce the following innerproduct on M atn,s (Fq ). At first, let n = 1 and ω1 = ω2 = (ξ100 , . . . , ξs00 ) ∈ M at1,s (Fq ). Then we put

(ξ10 , . . . , ξs0 ),

hω1 , ω2 i = hω2 , ω1 i =

(5)

s X

00 ξi0 ξs+1−i

i=1 (1)

(n)

(j)

Now, let Ωi = (ωi , . . . , ωi )T ∈ M atn,s (Fq ), i = 1, 2, ωi Then we put n X

hΩ1 , Ω2 i = hΩ2 , Ω1 i =

(6)

(j)

∈ M at1,s (Fq ), 1 ≤ j ≤ n.

(j)

hω1 , ω2 i

j=1

Notice that a specific choice of inner product (5) is very essential for our consideration. For example, it leads to the MacWilliams relations given in Theorems 3.1 and 3.2. Additionally, it was shown in (Theorem 4.1, [11]) that if C is a linear MDS code with respect to the ρ metric then C ⊥ (given by this innerproduct) is an MDS code as well. In general, this theorem fails for other choices of innerproducts. It is not known if there exist other innerproducts which may satisfy these theorems. Moreover, the relation between codes in this space and uniform distributions would also indicate that this is the natural innerproduct. For a given linear code C ⊂ M atn,s (Fq ) its dual code C ⊥ ⊂ M atn,s (Fq ) is defined by C ⊥ = {Ω2 ∈ M atn,s (Fq ) | hΩ2 , Ω1 i = 0 for all Ω1 ∈ C}.

(7)

It is obvious that C ⊥ is also a linear code, and (C ⊥ )⊥ = C.

(8)

Moreover the dimensions and cardinalities of the subspaces C and C ⊥ are related by (9)

d + d⊥ = ns, |C||C ⊥ | = q ns , |C| = q d , |C ⊥ | = q ns−d ,

where d is the dimension of C and d⊥ is the dimension of C ⊥ . MacWilliams-type theorems concern relations between weight enumerators for mutually dual linear codes. For example, in the case of s = 1 and arbitrary n and the ρ weight enumerator (4) satisfies the following relation (10)

W (C ⊥ | z) =

1 1−z (1 + (q − 1)z)n W (C | ) |C| 1 + (q − 1)z

by the classical MacWilliams theorem for the Hamming weight enumerators (cf. [7], Chapter 5, Theorem 13). 3

In the opposite case of n = 1 and arbitrary s the following identity was given in ([11], Theorem 4.4): (11)

(qz − 1)W (C ⊥ |z) + 1 − z = |C ⊥ |z s+1 (q(1 − z)W (C |

1 ) + qz − 1). qz

The main goal of the present paper is to extend MacWilliams-type theorems to the case of arbitrary n and s. However, the following counterexample shows that direct extensions do not exist for the ρ weight enumerator (4). Consider two linear codes C1 and C2 ⊂ M at2,2 (F2 ), 

(12)

 





 



0 0   1 0  0 0   0 0  C1 = { , }, C2 = { , } 0 0 1 0 0 0 0 1

Both codes have ρ weight enumerator (13)

1 + z2

Dual codes C1⊥ and C2⊥ ⊂ M at2,2 (F2 ) can be easily found: 

C1⊥ = {

 

 



0 0   0 1   1 1   1 1  , , , , 0 1 1 1 0 1 0 0  

 



 

 



0 1   1 0   0 0   1 0   } , , , 1 0 1 0 0 0 1 1 and 

C2⊥

 

 

 



0 0   1 0   0 1   0 0  = { , , , , 0 0 0 0 0 0 0 1  

 

 

 



1 1   1 0   0 1   1 1  , } , , 0 1 0 1 0 1 0 0

The ρ weight enumerator for C1⊥ and C2⊥ turns out to be different: (14)

W (C1⊥ | z) = 1 + 4z 4 + 2z + z 2 , W (C2⊥ | z) = 1 + 2z 4 + z 3 + 3z 2 + z

Therefore, the ρ weight enumerators (13) and (14) cannot be related by a MacWilliams type relation. One should conjecture that our inner product (5), (6) has to be replaced by a better one to obtain an appropriate MacWilliams-type theorem. However, such is not the case already for n = 1. Without going into a detailed consideration we shall compare innerproduct (5) with the common one: s (15)

[ω1 , ω2 ] =

X i=1

4

ξi0 ξi00 .

Consider two linear codes C1 and C2 ⊂ M at1,4 (F2 ), (16)

C1 = {0000, 1100, 1001, 0101}, C2 = {0000, 0100, 0001, 0101}.

Their duals with respect to innerproduct (5) are the following: (17)

C1⊥ = {0000, 0100, 1111, 1011}, C2⊥ = {0000, 0100, 0001, 0101}

Notice that four codes (16), (17) have the same ρ weight enumerators: (18)

W (Ci | z) = W (Ci⊥ | z) = 1 + z 2 + 2z 4 , i = 1, 2.

Denote by C1∗ and C2∗ codes dual to C1 and C2 with respect to inner product (15). We have (19) C1∗ = {0000, 0010, 1111, 1101}, C2∗ = {0000, 0010, 1000, 1010}. The ρ weight enumerators (4) for codes (19) are different: (20)

W (C1∗ | Z) = 1 + z 3 + 2z 4 , W (C2∗ | z) = 1 + z + 2z 3 .

Therefore, the ρ weight enumerators W (C | z) and W (C ∗ | z) cannot be related by a MacWilliams-type identity. The foregoing dramatic circumstances send us in search of more adequate definitions for weight enumerators. In the present paper, following an approach proposed in [11], Sec. 4.4, we consider orbits of a linear group preserving the weight ρ, and we show that weight enumerators associated with such orbits satisfy MacWilliams-type theorems for mutually dual linear codes. It turns out that the indicated group is transitive on each sphere for the metric ρ only in the two special cases of s = 1 or n = 1. In fact this observation gives an insight into why the ρ weight enumerators (4) satisfy MacWilliams-type identities only in the cited cases. It is worth noting that our treatment of the problem is close to a known approach of Delsarte (see [4], Chapter 6) to MacWilliams-type identities for the Hamming metric. There is also a significant relationship to [8], where they study ordered codes, ordered orthogonal arrays, (T, M, S)-nets and association schemes. In contrast, the approach in the present paper is motivated by the relationship of the codes with the Tsfasman-Rosenbloom metric to uniform distributions. The development of these theories arose independently, and we shall point out where equivalent results occur.

2

Geometry of the metric ρ

Consider balls and spheres in M atn,s (Fq ) with respect to the metric ρ, (21)

B (n,s) (r) = {Ω ∈ M atn,s (Fq ) | ρ(Ω) ≤ r}, 5

(22)

S (n,s) (r) = {Ω ∈ M atn,s (Fq ) | ρ(Ω) = r},

where 0 ≤ r ≤ ns is an integer. A detailed study of balls (21) and spheres (22) was given in ([11], Section 3.1). Here we state the corresponding results in a form adapted for the present consideration. At first, let n = 1. Then, using definitions (1) and (21), we find that (23)

B (1,s) (r) = {ω = (ξ1 , . . . , ξs ) ∈ M at1,s (Fq ) | ξi = 0 for i > r}

is a subspace of dimension r, and (24)

S (1,s) (0) = B (1,s) (0), S (1,s) (r) = B (1,s) (r) − B (1,s) (r − 1), r ≥ 1.

Notice that M at1,s (Fq ) splits into a disjoint union of spheres: (25)

M at1,s (Fq ) =

s [

S (1,s) (r).

r=0

For an arbitrary n we denote by Qn,s ⊂ Zn the following subset of integer vectors (26)

Qn,s = {R = (r1 , . . . , rn ) | 0 ≤ rj ≤ s, 1 ≤ j ≤ n}.

We regard the space M atn,s (Fq ) as a direct product of n copies of M at1,s (Fq ): (27)

M atn,s (Fq ) =

n Y

M at1,s (Fq ).

j=1

In so doing, we introduce subspaces VR ⊂ M atn,s (Fq ), (28)

VR =

n Y

B (1,s) (rj ), R ∈ Qn,s

j=1

with (29)

dim VR = r1 + . . . + rn ,

and subsets FR ⊂ M atn,s (Fq ), (30)

FR =

n Y

S (1,s) (rj ), R ∈ Qn,s .

j=1

The subsets (30) are said to be fragments. Notice that M atn,s (Fq ) splits into a disjoint union of fragments (30): (31)

M atn,s (Fq ) =

[

FR

R∈Qn,s

The following results were given in [11], Lemmas 3.1 and 3.3. 6

Lemma 2.1 Each ball (21) is a union of the following subspaces (28) (32)

B (n,s) (r) =

[

VR

r1 +...+rn =r

Each sphere (22) is a disjoint union of the following fragments (30) (33)

S (n,s) (r) =

[

FR

r1 +...+rn =r

Following ([11], Remark 2.2), we introduce groups Tns , Hn,s , with Tns ⊂ Hn,s of linear transformations on M atn,s (Fq ) preserving the weight ρ. At first, let n = 1, ω = (ξ1 , . . . , ξs ) ∈ M at1,s (Fq ), and Ts denote a group of all lower triangular s by s matrices over Fq with arbitrary nonzero diagonal elements. From definition (1) we immediately conclude that the mappings (34) τ : M at1,s (Fq ) 3 ω → ωτ ∈ M at1,s (Fq ), τ ∈ Ts preserve the weight ρ: ρ(ωτ ) = ρ(ω). Now, let Ω = (ω, . . . , ωn )T ∈ M atn,s (Fq ), ωj ∈ M at1,s (Fq ), 1 ≤ j ≤ n, and (35)

Tsn

=

n Y

Ts

j=1

denote a direct product of n copies of Ts . Then, it is obvious that the mappings t : M atn,s (Fq ) 3 Ω = (ω1 , . . . , ωn )T → Ωt = (ω1 τ1 , . . . , ωn τn )T ∈ M atn,s (Fq ), t = (τ1 , . . . , τn ) ∈ Tsn , (36) preserve the weight ρ : ρ(Ωt) = ρ(Ω). Furthermore, from (2) we conclude that the symmetric group Sn of all permutations of rows of the matrix Ω ∈ M atn,s (Fq ) preserves the weight ρ. Therefore, the direct product (37)

Hn,s = Sn × Tsn

forms a group of linear transformations on M atn,s (Fq ) preserving the weight ρ. We wish to describe orbits of groups Tsn and Hn,s on the space M atn,s (Fq ). For a given integer vector R = (r1 , . . . , rn ) ∈ Qn,s and a permutation σ ∈ Sn we write σR = (rσ(1) , . . . , rσ(n) ). The quotient set Qn,s /Sn can be identified with the following subset of integer vectors: (38)

Qn,s /Sn = {R = (r1 , . . . , rn ) | 0 ≤ r1 ≤ . . . ≤ rn ≤ s}.

For a given R = (r1 , . . . , rn ) ∈ Qn,s /Sn we introduce the stabilizing subgroup Sn(R) ⊂ Sn by (39)

Sn(R) = {σ ∈ Sn | σR = R} 7

We also define the following non-negative integers (40)

nm = nm (R) = |{i | ri = m}|, 0 ≤ m ≤ s

It is obvious that n0 + n1 + . . . + ns = n and the subgroup Sn(R) is isomorphic to the direct product Sn0 × Sn1 × . . . × Sns . Therefore, the number of cosets Sn /Sn(R) is equal to the polynomial coefficient   n! n  = (41) n0 !n1 ! . . . ns ! n0 , n1 , . . . , ns Introduce the following disjoint unions of fragments (30) (42)

ΦR =

[

FσR ,

R ∈ Qn,s /Sn .

(R) σ∈Sn /Sn

where σ runs over a fixed set of representatives for the cosets Sn /Sn(R) . Proposition 2.2 (i) T -orbits of the group Tsn on M atn,s (Fq ) coincide with fragments (30). Hence, there is a bijection between T -orbits and the set Qn,s . (ii) H-orbits of the group Hn,s on M atn,s (Fq ) coincide with unions of fragments of the form (42). Hence, there is a bijection between H-orbits and the quotient set Qn,s /Sn . Proof. (i) At first, let n = 1. From (23), (24) and (30) we conclude that a fragment Fr = S (1,s) (r), 0 ≤ r ≤ s, consists of all ω = (ξ1 , . . . , ξs ) ∈ M at1,s (Fq ) such that ξi = 0 for i > r and ξr 6= 0. Now it can be easily checked that the group Ts is transitive on each fragment S (1,s) (r). Obviously, the foregoing implies (i) for n = 1 and for an arbitrary n as well, because all the subjects M atn,s (Fq ), FR , Hsn are direct products of n copies of those for n = 1. (ii) In view of definitions (37) and (42) statement (ii) follows at once from the statement (i). 2 Comparing Proposition 2.2 with Lemma 2.1, we immediately obtain the following: Corollary 2.3 The group Hn,s is transitive on each sphere S (n,s) (r), 0 ≤ r ≤ ns, only in the two special cases of s = 1 or n = 1. As we mentioned above in the Introduction this circumstance explains why MacWilliamstype identities do not exist for the ρ weight enumerators (4) in the case of s > 1 and n > 1. Consider a semidirect product Gn,s , of the group Hn,s and the group of translations of M atn,s (Fq ). The group Gn,s preserves the ρ metric: ρ(gΩ1 − gΩ2 ) = ρ(Ω1 − Ω2 ), Ω1 , Ω2 ∈ M atn,s (Fq ), g ∈ Gn,s . 8

The corresponding action g : (Ω1 , Ω2 ) → (gΩ1 , gΩ2 ), g ∈ Gn,s , splits the set of pairs (Ω1 , Ω2 ) ∈ M atn,s (Fq ) × M atn,s (Fq ) into finitely many G-orbits. It can be easily checked that each G-orbit consists of pairs (Ω1 , Ω2 ) such that Ω1 − Ω2 belong to an H-orbit in M atn,s (Fq )

3

MacWilliams identities for enumerators associated with orbits

For a given linear code C ⊂ M atn,s (Fq ) the following sets of nonnegative integers tR (C) = |{C ∩ FR }|, R ∈ Qn,s ,

(43) and (44)

hR (C) = |{C ∩ ΦR }| =

X

tσR (C), R ∈ Qn,s /Sn ,

σ∈Sn

are respectively called the T - and H- spectra of the code C. Obviously, quantities (43) and (44) are equal to the number of points of C falling into the corresponding orbits of the group Tsn and Hn,s . For the ρ weight spectrum (43) we have the relation (45)

wr (C) =

X

X

tR (C) =

r1 +...+rn =r

hR (C)

0 ≤ r1 ≤ . . . , rn ≤ s r1 + . . . + rn = r

by Lemma (2.1) and Proposition (2.2). Define the T -enumerator by T (C | Z1 , . . . , Zn ) =

(46)

X

tR (C)

R∈Qn,s

n Y

zr(j) , j

j=1

(j)

where Zj = (z0 , . . . , zs(j) )T ∈ Cs+1 , 1 ≤ j ≤ n, are n complex vectors with s + 1 components. In another form, (46) can be written as (47)

T (C | Z1 , . . . , Zn ) =

X

Υ(Ω | Z1 , . . . , Zn )

Ω∈C

where Υ(Ω) = za(1) za(2) . . . za(n) and ρ(ωi ) = ai , 1 ≤ i ≤ n. n 1 2 Similarly, we define the H-enumerator by 9

H(C | Z) =

(48)

X

hR (C)

R∈Qn,s /Sn

n Y

zrj ,

j=1

where Z = (z0 , . . . , zs )T ∈ Cs+1 . A weight enumerator equivalent to the H-enumerator defined in terms of association schemes was given in [8]. Notice that enumerator (46) is a polynomial of degree at most one in each of n(s + 1) (j) variables zi , 0 ≤ i ≤ s 1 ≤ j ≤ n, while enumerator (48) has degree at most n in each of s + 1 variables zi , 0 ≤ i ≤ s. Comparing (46) and (48), we obtain the equality H(C | Z) = T (C | Z, Z, . . . , Z).

(49)

Similarly, from (4), (45) and (48) we find that W (C | z) = H(C | 1, z, . . . , z s ).

(50)

Introduce a linear transformation Θs : Cs+1 → Cs+1 by setting Z 0 = Θs Z,

(51) where

(52)z00 = z0 + (q − 1)z1 + q(q − 1)z2 + q 2 (q − 1)z3 + . . . + q s−2 (q − 1)zs−1 + q s−1 (q − 1)zs z10 = z0 + (q − 1)z1 + q(q − 1) + q 2 (q − 1)z3 + . . . + q s−2 (q − 1)zs−1 + −q s−1 zs . . . . 0 zs−2 0 zs−1 zs0

= z0 + (q − 1)z1 + q(q − 1) − q 2 z3 = z0 + (q − 1)z1 − qz2 = z0 − z1

Notice that writing relations (52), we assume that Z = (z0 , z1 , z2 , . . .) is an infinite sequence with zi = 0 for i > s. Thus in (51) the s + 1 by s + 1 matrix Θs = ||θlk ||, 0 ≤ l, k ≤ s, has the following entries 10

(53)

      

1 q (q − 1) θlk =  −q l−1      0 l−1

if l = 0, if 0 < l ≤ s − k, if l + k = s + 1, if l + k > s + 1.

This matrix is given as the first eigenmatrix of an association scheme in [8]. For example, we have for s = 1, 2 and 3. 



1 q−1  Θ1 =  1 −1 



1 q − 1 q(q − 1)     Θ2 =  1 q − 1 −q  1 −1 0

(54)



Θ3 =

     

1 q − 1 q(q − 1) q 2 (q − 1) 1 q − 1 q(q − 1) −q 2 1 q−1 −q 0 1 −1 0 0

      

Notice that Θs satisfies the equality (55)

{Θs }2 = q 2 Is ,

where Is denotes the identity matrix of size s + 1. Equality (55) can be proven by a direct calculation, but we shall derive it from Theorem 3.1 given below. Equation (55) is equivalent to Lemma 1.5 in [8]. Now we are in position to state our results on MacWilliams-type identities for T - and H-enumerators. Theorem 3.1 The T -enumerators of mutually dual linear codes C, C ⊥ ⊂ M atn,s (Fq ) are related by 1 (56) T (C | Θs Z1 , . . . , Θs Zn ). T (C ⊥ | Z1 , . . . , Zn ) = |C| Notice that (56) together with (8) and (9) immediately implies equality (55). Using (49), from Theorem 3.1 we derive the following: Theorem 3.2 The H-enumerator of mutually dual linear codes C, C ⊥ ⊂ M atn,s (Fq ) are related by 1 (57) H(C ⊥ |Z) = H(C | Θs Z) |C|

11

It should be mentioned that relations of the type (57) arise in the context of association schemes, (cf. Godsil [6]), see also Martin and Stinson [8] for applications to ordered orthogonal arrays. Namely, Proposition 3.2 in [8] is equivalent to Theorem 3.2, however the notation and setting are different and it is proven in a different manner so we include the result here. Using (50), one can easily show that in the special cases of s = 1 and n = 1 relation (57) implies identities (10) and (11), respectively. The necessary simple calculations are left to the reader. Among direct corollaries of the above results we should mention the following: Theorem 3.3 Suppose that two linear codes C1 and C2 ⊂ M atn,s (Fq ) have the same T spectra (43) (or respectively H-spectra, (44)), then T -spectra (and, respectively H-spectra) of their dual codes C1⊥ and C2⊥ coincide. In the next section we shall see that Theorem 3.3 can essentially be improved. Namely, we shall give explicit relations between T -spectra (and, respectively H-spectra) of mutually dual codes. Turning to the proof of Theorem 3.1 we recall necessary facts on the Fourier transform on vector spaces over finite fields. For details we refer to [7], Chapter 5. Introduce a function Ψ(Ω1 , Ω2 ), Ω1 , Ω2 ∈ M atn,s (Fq ) by setting (58)

√ 2π −1)T rhΩ1 , Ω2 i ), Ψ(Ω1 , Ω2 ) = exp( p

where e−1

T rξ = ξ + ξ p + . . . ξ p

denotes the trace of an element ξ ∈ Fq , q = pe , over a simple field Fp , and h·, ·i is the inner product (6). It is obvious that for each Ω the function Ψ(Ω, ·) is an additive character on the vector space M atn,s (Fq ) : (59) Ψ(Ω, Ω1 + Ω2 ) = Ψ(Ω, Ω1 )Ψ(Ω, Ω2 ). Furthermore, each character of the additive group of M atn,s (Fq ) coincides with Ψ(Ω, ·) for a suitable Ω. The Fourier transform fb of a given complex function f : M atn,s (Fq ) → C is defined by (60)

fb(Ω1 ) =

X

Ψ(Ω1 , Ω2 )f (Ω2 )

Ω2 ∈M atn,s (Fq )

The following results are well known in the theory of abelian groups, see [7] for example. 12

Lemma 3.4 Let D, D⊥ ⊂ M atn,s (Fq ) be subspaces mutually dual with respect to the inner product h·, ·i. Then (i) One has the relation (61)

X

Ψ(Ω1 , Ω2 ) =

Ω2 ∈D

 

if Ω1 ∈ D⊥ , if Ω1 ∈ / D⊥ .

|D|,  0,

(ii) One has the Poisson summation formula X

(62)

f (Ω) =

Ω∈D⊥

1 X b f (Ω) |D| Ω∈D

for an arbitrary function f . We write χ(E | ·) for the indicator function of a given subset E ⊂ M atn,s (Fq ), χ(E | Ω) =

(63)

 

1, 0,



if Ω ∈ E if Ω ∈ / E.

We need the following result concerning specific subspaces (28) (cf. [11], Lemma 4.2). Lemma 3.5 For a given r = (r1 , . . . , rn ) ∈ Qn,s cf. (26) define R∗ = (r1∗ , . . . , rn∗ ) ∈ Qn,s by rj∗ = s − rj , 1 ≤ j ≤ n. Then, one has the relations (64)

VR⊥ = VR∗

(65)

b R | Ω) = q r1 +...+rn χ(VR∗ |Ω). χ(V

Proof. Relation (64) follows at once from definitions (7), (23), and (28) by a simple calculation. Relation (65) directly follows from (64), (61) and (9). 2 Proof of Theorem 3.1: Consider a fragment (30). We have χ(FR | Ω) =

(66)

n Y

χ(S (1,s) (rj ) | ωj ).

j=1

Introduce the following functions f (Ω | Z1 , . . . , Zn ) =

(67)

n Y

f (ωj | zj ),

j=1

f (ω | Z) =

(68)

s X

χ(S (1,s) (r) | ω)zr .

r=0 T

Here Ω = (ω1 , . . . , ωn ) ∈ M atn,s (Fq ), ωj ∈ M at1,s (Fq ), and Z = (z0 , . . . , zs ), Zj = ∈ Cs+1 , 1 ≤ j ≤ n.

(j) (z0 , . . . , zs(j) )

13

Comparing (66), (67), and (68), we find that (69)

X

f (Ω | Z1 , . . . , Zn ) =

χ(FR | Ω)

n Y

zr(j) j

j=1

R∈Qn,s

Therefore from (46) and (69) we obtain the relation (70)

T (C ⊥ | Z1 , . . . , Zn ) =

X

f (Ω | Z1 , . . . , Zn )

Ω∈C ⊥

We wish to apply the Poisson summation formula (62) to (70). For this purpose we need to evaluate the Fourier transform (60) of functions (67), (68). First of all, using (24) and (68), we find that (71)

f (ω | z0 , z1 , . . . , zs ) = χ(B (1,s) (0) | ω)z0 + (χ(B (1,s) (1) | ω) − χ(B (1,s) (0) | ω)z1 + (χ(B (1,s) (2) | ω) − χ(B (1,s) (1) | ω)z2 + . . . + (χ(B (1,s) (s) | ω) − χ(B (1,s) (s − 1) | ω)zs = χ(V0 |ω)(z0 − z1 ) + χ(V1 |ω)(z1 − z2 ) + . . . + χ(Vs−1 |ω)(zs−1 − zs ) + χ(Vs |ω)zs ,

where Vr = B (1,s) (r), 0 ≤ r ≤ s, are subspaces (28) for n = 1. Now, by (66) we have (72)

fb(ω | z0 , z1 , . . . , zs ) = χ(Vs |ω)(z0 − z1 ) + χ(Vs−1 | ω)q(z1 − z2 ) + . . . + χ(V1 | ω)q s−1 (zs−1 (zs−1 − zs ) + χ(V0 |ω)q s zs = χ(V0 | ω)(z00 − z10 ) + χ(V1 | ω)(z10 − z20 ) + . . . 0 + χ(vs−1 | ω)(zs−1 − zs0 ) + χ(vs ω)zs0

= f (ω | z00 , z10 , . . . , zs0 ), where (73)

zs0 = z0 − z1 , 0 − zs0 = q(z1 − z2 ), zs−1 0 0 zs−2 − zs−1 = q 2 (z2 − z3 ),

. . 14

. z10 − z20 = q s−1 (zs−1 − zs ), z00 − z10 = q s zs .

Solving equations (73) relative to variables zi0 , 0 ≤ i ≤ s, we immediately obtain relations (52). Hence, (74) fb(ω | Z) = f (ω | Θs Z). Therefore, by (67) and (74) we have the relation (75)

fb(Ω | Z1 , . . . Zn ) = f (Ω | Θs Z1 , . . . , Θs Zn ).

Using (75) and applying the Poisson summation formula (70) we obtain the relation in question (56). The proof of Theorem 3.1 is complete. 2

4

Relations between T - and H-spectra for mutually dual codes

In the context of the above Theorem 3.1 the following question naturally arises: Suppose we know the T -spectrum (or, respectively the H-spectrum) of a given linear code C ⊂ M atn,s (Fq ), how the T -spectrum (and, respectively, the H-spectrum) of the dual code C ⊥ ⊂ M atn,s (Fq ) can be determined? In the present section we give explicit formulas which solve the question. Notice that in the special case of s = 1 such formulas are well known, and their construction involves the Krawtchouk polynomials (see [7], Chapter 5). In the case of arbitrary s our formulas involve generalizations of these subjects. Introduce the following coefficients: 

(76)







n A  a1 , a2 , . . . , an  Y Ls  = Ls  = θaj ,bj , b1 , b2 , . . . , bn B j=1

where A = (a1 , . . . , an ), B = (b1 , . . . , bn ) ∈ Qn,s (cf. (26), and θl,k , 0 ≤ l, k ≤ s, are entries (53) of the matrix Θs . It is obvious that quantities (76) are coefficients in the following expression: (77)

n X s Y

(

j=1 i=0



(j) θaj ,i zi )



n A Y (j) = Ls  zbj B j=1 B∈Qn,s

X

15

Theorem 4.1 The T -spectra (43) of mutually dual linear codes C, C ⊥ ⊂ M atn,s (Fq ) are related by   X 1 A  tA (C). (78) Ls  tB (C ⊥ ) = |C| A∈Qn,s B Proof. It suffices to compare expansion(77) with the MacWilliams-type identity (56). 2 Thus, the (s + 1)n -dimensional vectors {tA (C ⊥ ), A ∈ Qn,s } and {tB (C ⊥ ), B ∈ Qn,s } are related by the n-linear mapping ΘTs ⊗ . . . ⊗ ΘTs = ⊗nj=1 ΘTs . Notice that (78) together with (8) and (9) immediately implies the following relation 

(79)







R   A  Ls  Ls = q ns δA,B , B R R∈Qn,s X

where δA,B denotes the Kroneker symbol. In fact, relation (79) is equivalent to equation (55). If Z1 = . . . = Zn = Z, relation (77) takes the form (80)





n A Y Ls  zb ( θaj ,i zi ) = B j=1 j j=1 i=0 B∈Qn,s n X s Y

X

Notice that products in both sides of (80) are invariant under arbitrary permutations A = (a1 , . . . , an ) → σA = (aσ(1) , . . . , aσ(n) ) and B = (b1 , . . . , bn ) → λB = (bλ(1) , . . . , bλ(n) ) σ, λ ∈ Sn . Therefore, (81)



n X s Y



n A Y ( θaj ,i zi ) = Ks  zb B j=1 j j=1 i=0 B∈Qn,s /Sn

X

where 







A  a1 , . . . , a n  = Ks  = Ks  B b1 , . . . , b n 



σA  Ls  = B σ∈Sn X

n X Y σ∈Sn j=1

θaσ(j) ,bj =





A  Ln  = σB σ∈Sn X

n X Y σ∈Sn j=1

16

θaj ,bσ(j) .

Hence coefficients (82) satisfy the relation 











σA  A  A  Ks  = Ks  = Ks  , B σB B

(82)

and therefore they are determined completely by their values for A, B ∈ Qn,s /Sn . Using definitions (39), (40), (41), we can write (82) in the following form 







A   n  Ks  = B n0 (A), . . . , ns (A)

(83)





X

σ∈Sn /Sn





n   n0 (B), . . . , ns (B)



σA  Ls  = B (A)

X



A  Ls  σB (B)

σ∈Sn /Sn

It can be easily checked that in terms of coefficients (82) relation (79) takes the form 







A  R  = q ns δA,B . Ks  Ks  R B R∈Qn,s /Sn X

(84)

Now, our result on H-spectra can be stated as follows: Theorem 4.2 The H-spectra (44) of mutually dual linear codes C, C ⊥ ⊂ M atn,s (Fq ) are related by   X 1 A  hA (C). hB (C ⊥ ) = (85) Ks  |C| A∈Qn,s /Sn B Proof. It suffices to compare expansion (80) with the MacWilliams-type identity (57). 2 In the special case of s = 1 coefficients (82) coincide with the Krawtchouk polynomials. Indeed, from (38) we conclude that for s = 1 integer vectors A, B ∈ Qn,s /Sn have the form A = (0, . . . , 0, 1, . . . , 1) where there are l ones and B = (0, . . . , 0, 1, . . . , 1) where there are k ones. Hence we can put,   A   = Pk (l), 0 ≤ k, l ≤ n. (86) B Since





1 q−1  , Θ1 =  1 −1 see (54), we have (87)

n X 1 Y

(

j=1 i=0

1 X

θaj ,i zi ) = (

i=0

1 X

θ0,i zi )n−l (

θ1,i zi )l = (z0 + (q − 1)z1 )n−l (z0 − z1 )l

i=0

17

and

n Y

zbj = z0n−k z1k .

j=1

Therefore, relation (80) takes the form (z0 + (q − 1)z1 )n−l (z0 − z1 )l =

n X

Pk (l)z0n−k z1k ,

k=0

that coincides with the known definition for the Krawtchouk polynomials Pk (l), cf. [7], Eq. (5.54). It is well known, cf. [7], Eq. (5.59) that the Krawtchouk polynomials Pk (l) satisfy the relations     n n (q − 1)l   Pk (l) = (q − 1)k   Pl (k) (88) l k 



n n! denotes the usual binomial coefficients. for all 0 ≤ k, l ≤ n. Here   = k!(n−k)! k Relations (88) together with (84) imply the following, cf. [7], Eq. (5.28). (89)

n X









n n (q − 1)i   Pl (i)Pk (i) = q n   δl,k . i l i=0

Thus, the Krawtchouk polynomials Pk (i) are orthogonal with respect to the discrete measure   i n  Mi = (q − 1) i concentrated on the set of n + 1 points {0, , . . . , n}. 







A  A  We wish to give similar results for coefficients Ls  and Ks  in the case of B B arbitrary s. A careful consideration of entries (53) leads to the following relations q l θlk = q k θkl for 0 < l, k ≤ s, and θ0k = q k−1 (q − 1)θk0 for 0 < k ≤ s. Therefore, for all 0 ≤ l, k ≤ s we have (90)

vl θlk = vk θkl ,

18

where (91)

vl =

 

ql if 0 < l ≤ s,  q(q − 1)−1 if l = 0

For an arbitrary integer vector R = (r1 , . . . , rn ) ∈ Qn,s we put vR =

(92)

n Y

vrj

j=1

Notice that (93)

vσR = vR , σ ∈ Sn .

With notations (40), (41), and (92), we have the following result. Theorem 4.3 (i) For all A, B ∈ Qn,s one has the relations 

(94)

vA Ls 

and







A  B  = vB Ls  B A









R   R  vR Ls  Ls = q ns vA δA,B A B R∈Qn,s X

(95)

(ii) For all A, B ∈ Qn,s /Sn one has the relations 

(96)



and (97)













B  n n A    vB Ks    vA Ks  = A n0 (B), . . . , ns (B) n0 (A), . . . , ns (A) B 











n R  R    vR Ks  Ks  = n0 (R), . . . , ns (R) A B R∈Qn,s /Sn X





n   q ns vA δA,B . n0 (A), . . . , ns (A) Proof. Relations (94) follow at once from definition (76) and relations (90) and (92). Substituting (94) to equality (77), we obtain relation (95). Similarly, relations (95) follow from relations (83), (93), and (94). Substituting (96) to equality (84), we obtain relation (97). 2 







R  R  Thus, coefficients Ls  and Ks  as functions of R are orthogonal relative to A A 

discrete measures vR , R ∈ Qn,s , and respectively 

19



n  vR , R ∈ Qn,s /Sn . n0 (R), . . . , ns (R)

Notice that in view of (86) relations (96) and (97) for s = 1 coincide with the corresponding relations (88) and (89) for the Krawtchouk polynomials. It should be mentioned that the Krawtchouk polynomials naturally arise as zonal spherical functions for the so called hyperoctahedral group. This circumstance has important corollaries in coding theory. We refer to Conway and Sloane [3] for details. It seems likely that coefficients (76) and (82) could be interpreted in terms of zonal spherical functions for the groups Tsn and Hn,s , respectively. Further consideration of these questions should be very interesting.

References [1] E. Bannai and T. Ito, “Algebraic Combinatorics I. Association Schemes”, The Benjamin Cummings Publ. Com., London, 1984. [2] W.W.L. Chen and M.M. Skriganov, Explicit Constructions in the Classical Mean Square Problem in Irregularities of Point Distributions, Macquarie University and Steklov Math. Inst. at St. Petersburg, preprint, 1999. [3] J.H. Conway and N.J.A. Sloane, “Sphere Packing, Lattices and Groups (2nd ed.)”, Springer-Verlag, New York, 1993. [4] P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Research Reports Supplements, No. 10, 1973. [5] S.T. Dougherty and M.M. Skriganov, MacWilliams-type Theorems for a Non-Hamming Metric, University of Scranton and Steklov Math. Inst. at St. Petersburg, preprint, 2000. [6] C.D. Godsil, MacWilliams theorems for product schemes, preprint. [7] F.J. MacWilliams and N.J.A. Sloane, “The Theory of Error-Correcting Codes”, NorthHolland, Amsterdam 1977. [8] W.J. Martin and D.R. Stinson, Association schemes for ordered orthogonal arrays and (T, M, S)-nets, Canad. J. Math. Vol 51, 326-346, 1999. [9] M. Yu Rosenbloom and M. A. Tsfasman, Codes for the m-metric, Problems of Information Transmission, Vol. 33, No. 1, 45-52, 1997. (Translated from Problemy Peredachi Informatsii, Vol 33, No. 1, 55-63, 1996.) [10] M.M. Skriganov, Uniform Distributions, Error-Correcting Codes, and Interpolations over Finite Fields preprint, 1998.

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[11] M.M. Skriganov, Coding theory and uniform distributions, Algebra i Analiz, Vol 13, No. 2, 170-221, 2001. (Translation to appear in St. Petersburg Math. J.)

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