In this section we describe how the previous results generalize to MacWilliams identities for codes in Rn1×Rn2, where one considers different induced partitions
on Rn1and Rn2. The ideas generalize straightforwardly to any finite number of factors.
Several of such cases for split enumerators have been investigated in the literature. They are discussed below. The main idea of this section has been used already in [18, Ch. 5, § 6] for the split Hamming weight enumerator.
SupposeP and Q are partitions of R. Moreover, let ˜P and ˜Q be induced partitions
on Rn1and Rn2 in the sense of Definition 7, respectively. Thus, either one may be the
induced product or symmetrized partition. Write ˜P = (Pl)Ll=1and ˜Q = (Qm)Mm=1, and
for v∈ Rn1 (resp. v∈ Rn2) denote by[v] the index of the partition set in ˜P (resp. ˜Q)
containing v. Let the resulting enumerating functions for ˜P and ˜Q take values in
C[Z] and C[T], respectively, where Z and T are appropriate lists of indeterminates;
see Theorems 8 and 9. For any v∈ Rn1 let Z
[v] denote the monomial associated
with the partition set P[v] and similarly for the vectors in Rn2. Then the partition
enumerators are given by∑v∈CZ[v]for codesC in Rn1 and∑
v∈CT[v]for codes in Rn2.
Definition 18. The split partition enumerator of a codeC ⊆ Rn1×Rn2with respect to
the partition ˜P × ˜Q is defined as SPEP× ˜˜ Q,C= ∑
(v,w)∈C
Z[v]T[w]∈ C[Z, T].
The coefficient of a monomial ZlTmequals the cardinality∣C ∩Pl×Qm∣.
It is not hard to see [28, Thm. 3] that if bothP and Q are F-partitions on R, then
˜
P × ˜Q is an F-partition on Rn1× Rn2. As a consequence, there is a MacWilliams
identity between the split partition enumerators of a code and its dual. We can make this identity precise by combining the previous results.
From Theorems 8 and 9 we know that there exist transformsM1∶ C[Z] → C[Z] and
M2∶ C[T] → C[T] such that PEP,C˜ ⊥ 1 = 1 ∣C1∣M 1(PEP,C˜ 1) and PEQ,C˜ ⊥ 2 = 1 ∣C2∣M 2(PEQ,C˜ 2)
for all codesCi⊆ Rni, i= 1,2. Now we can formulate
Theorem 19. Define the transform ¯M ∶ C[Z,T] → C[Z,T] as the C-algebra ho-
momorphism given byZlTm↦ M1(Zl)M2(Tm). Then we have the MacWilliams
identity
SPEP× ˜˜ Q,C⊥=
1
∣C∣M(SPE¯ P× ˜˜ Q,C)
for each codeC ⊆ Rn1×Rn2.
Sketch of Proof: We start again with the MacWilliams identity for the full weight
enumerator given in Theorem 6. From the proofs of Theorems 8 and 9 we know that
φi○M = Mi○φifor i= 1,2, where M is as in Theorem 6 and where
are given by φ1(Xv) = Z[v]and φ2(Xw) = T[w]. Furthermore, define
φ∶ C[X(v,w)∣ (v,w) ∈ Rn1×Rn2] Ð→ C[Z, T], X
(v,w)z→ Z[v]T[w]. Using the group homomorphism property of the character χ it is not hard to see that
φ(M(X(v,w)) = M1(φ1(Xv))M2(φ2(Xw)) = ¯M○φ(X(v,w)).
Now the identity in (4) along with SPEP× ˜˜ Q,C= φ(fweC) leads to the desired identity.
◻
This result covers several results known from the literature.
Example 20. (1) Consider the Hamming weight on both Rn1 and Rn2. In this case
we obtain the simple the identity SPEP× ˜˜ Q,C⊥(W0,W1,Z0,Z1)
=∣C∣1 SPEP× ˜˜ Q,C(W0+(∣R∣−1)W1,W0−W1,Z0+(∣R∣−1)Z1,Z0−Z1), (12)
where the coefficient of W1iWn1−i
0 Z j 1Z
n2− j
0 equals the number of codewords having
Hamming weight i on the first n1coordinates and Hamming weight j on the last n2
coordinates. This identity has been derived already by MacWilliams and Sloane for codes over the binary field in [18, Ch. 5, Eq. (52)] and by Simonis [22, Eq. (3’)], where the codewords are divided into t blocks of coordinates. A similar identity can
be found in [4] by El-Khamy and McEliece. The latter authors also observe that ifC
is a systematic[n,k] code, then the split weight enumerator for the Hamming weight
on both parts is the input-redundancy weight enumerator which keeps track of the input weights in combination with the corresponding redundancy weight. This allows them to apply their identity to MDS codes in order to derive further results on the bit error probability for systematic RS codes. In [16] this weight enumerator has been used to derive a MacWilliams identity for the input-output weight enumerators of direct-product single-partity-check codes.
(2) One should note that the support-tracker discussed in Example 16 as well as the Product Lee Weight Enumerator in Example 15 are special cases of the split weight enumerator, where we partition the codewords into n blocks of length 1.
We close this note with the Rosenbloom-Tsfasman weight for matrix codes. The following result can also be found in [3] by Dougherty and Skriganov.
Example 21. Recall the Rosenbloom-Tsfasman metric ρ from Example 17. Consider the vector spaceFs×nof all s×n-matrices over the field F = Fq. Denote the rows of
a matrix M∈ Fs×nby M1,...,Ms∈ Fn. For r∶= (r1,...,rs) ∈ N ∶= {0,...,n}sdefine
the set Pr∶= {M ∈ Fs×n∣ ρ(Mi) = ri, i= 1,...,s}. Then (Pr)r∈N forms a partition on
Fs×n. It is the direct product of the Rosenbloom-Tsfasman partition onFnextended toFn×...×Fn. Define the split weight enumerator of a codeC ⊆ Fs×nwith respect to the RT-metric on each factorFnas
SPEC∶= ∑ M∈C s ∏ i=1 Zi,ρ(Mi)∈ C[Z1,0,...,Z1,n,...,Zs,0,...,Zs,n].
Festschrift in Honor of Uwe Helmke H. Gluesing-Luerssen Then SPEC is a homogeneous polynomial of degree s, and for r∶= (r1,...,rs) ∈ N
the coefficient of∏si=1Zi,riequals the number of codewords inC ∩Pr. In [3, Sec. 3]
this enumerator has been coined the T-enumerator. For a codeC ⊆ Fs×nwe define the reversed dual as
C⍊∶= {B ∈ Fs×n∣ ∑s
i=1⟨Bi,AiJ⟩ = 0 for all A ∈ C},
where the matrix J is as in (10); see also [3, p. 83]. The same line of reasoning as in the proof of Theorem 19 along with the MacWilliams identity derived in Example 17 leads to the identity
SPEC⍊=
1
∣C∣M′(SPEC), where M′(∏si=1Zi,ri) = ∏
s
i=1(Zi,0+ ∑nj=1−riqj−1(q − 1)Zi, j− qn−riZi,n−ri+1), see (11).
This reproduces Theorem 3.1 in [3].
It is worth mentioning that the cumulative Rosenbloom-Tsfasman weight ρ(M) ∶=
∑s
i=1ρ(Mi) does not satisfy a MacWilliams identity. In [3] a pair of codes with the
same cumulative Rosenbloom-Tsfasman weight enumerator are given, and where the reversed dual codes have different enumerators.
Acknowledgments
The author was partially supported by National Science Foundation grant #DMS- 0908379.
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