A partition on R induces two specific partitions on Rn: the product partition and
the symmetrized partition. Both partitions give naturally rise to enumerators. We show that if the partition on R is an F-partition, then both these enumerators satisfy a MacWilliams identity. Examples will be presented in the next section.
LetP = {P1,...,PL} be an F-partition on R. For α ∈ R denote by [α] ∶= [α]P the
index of the partition set containing α∈ R.
Definition 7. (a) The induced product partition of Rnis defined as Pn∶= (P
l1×...×Pln)(l1,...,ln)∈{1,...,L}n.
(b) The composition vector of v= (v1,...,vn) ∈ Rnis defined as
compP(v) = (s1,...,sL), where sl= ∣{t ∣ vt∈ Pl}∣.
It is contained in the set S ∶= {(s1,...,sL) ∈ NL0∣ ∑Ll=1sl= n}. The induced sym-
metrized partitionon Rnis defined as Pn
Festschrift in Honor of Uwe Helmke H. Gluesing-Luerssen Note that the partition sets in the product partitionPncollect all vectors for which each entry is contained in a prescribed partition set, whereas the sets in the sym- metrized partition contain all vectors that have the same number of entries (disregard- ing position) in a given partition set.
For a given codeC ⊆ Rnwe may now define two types of partition enumerators. The
following two results show that both of them satisfy a MacWilliams identity. We start with the product partition. Recall the notation[α] for α ∈ R.
Theorem 8. LetC ⊆ Rnbe a code. The polynomialPEPn,C∶= ∑v∈C∏ni=1Yt,[v t], con-
tained in the polynomial ring
̃V ∶= C[Yt, j∣ t = 1,...,n, j = 1,...,L],
is called theproduct partition enumerator ofC with respect to P. The coefficient of ∏n
t=1Yt,lt equals the cardinality ofC ∩(Pl1×...×Pln). The product partition enumera-
tor satisfies the MacWilliams Identity PEPn,C⊥=
1
∣C∣M(PẼ Pn,C), (5)
where the MacWilliams transform ̃M ∶ ̃V Ð→ ̃V is defined as the algebra homo-
morphism satisfying ̃M(Yt,[α]) = ∑β∈Rχ(αβ)Yt,[β]= ∑ L
l=1∑β∈Plχ(αβ)Yt,l for all
t= 1,...,n and α ∈ R. In particular, ̃M is well-defined.
Proof. First of all, notice that∑β∈Rχ(αβ)Yt,[β]= ∑Ll=1∑β∈Plχ(αβ)Yt,l. By Defini-
tion 4, the coefficient∑β∈P
lχ(αβ) does not depend on the choice of α in its partition
set P[α], and this establishes the well-definedness of ̃M.
Consider now the situation of Theorem 6, and let φ∶ C[Xv∣ v ∈ Rn] Ð→ ̃V be the
substitution homomorphism defined via φ(Xv) = ∏nt=1Yt,[vt]. Using the group homo-
morphism property of the character χ one computes
φ○M(Xv) = ∑ w∈Rn χ(⟨v,w⟩) n ∏ t=1 Yt,[wt]= n ∏ t=1β∑∈R χ(vtβ)Yt,[β]= ̃M○φ(Xv).
Now Theorem 6 implies PEPn,C⊥= φ(fweC⊥) = 1
∣C∣φ○M(fweC) =∣C∣1M(PẼ Pn,C),
as desired.
Notice that we may write the identity (5) in the form PEPn,C⊥(Yt,l∣ t = 1,...,n, l = 1,...,L) =
1
∣C∣PEPn,C(KYt∣ t = 1,...,n),
where Yt= (Yt,1,...,Yt,L)Tand K= (km,l) ∈ CL×L is the Krawtchouk matrix of the
partitionP with entries defined in (3).
Theorem 9. For a codeC ⊆ Rnwe define thesymmetrized partition enumerator with
respect toP as PEPn
sym,C∶= ∑v∈C∏
n
t=1Z[vt]. It is a homogeneous polynomial of de-
gree n in the polynomial ring ̂V∶= C[Zj∣ j = 1,...,L]. The coefficient of the monomial
∏L j=1Z
sj
j inPEPsymn ,Cequals the cardinality∣{v ∈ C ∣ compP(v) = (s1,...,sL)}∣. The
symmetrized partition enumerator satisfies the MacWilliams Identity PEPn
sym,C⊥=
1
∣C∣M(PÊ Pn
sym,C), (6)
where the MacWilliams transform ̂M ∶ ̂V Ð→ ̂V is the (well-defined) algebra ho-
momorphism given by ̂M(Z[α]) = ∑β∈Rχ(αβ)Z[β]= ∑Ll=1∑β∈Plχ(αβ)Zl for all
α∈ R.
Proof. Again, the well-definedness of ̂M follows from the fact that P is an F-
partition on R. Let ̃V and ̃M be as in Theorem 8 and consider the substitution
homomorphism ψ∶ ̃V Ð→ ̂V given by ψ(Yt, j) = Zj. Then
ψ○ ̃M(Yt,[α]) = ψ( ∑ β∈R
χ(αβ)Yt,[β]) = ∑ β∈R
χ(αβ)Z[β]= ̂M○ψ(Yt,[α])
for all Yt,[α]. Now Theorem 8 yields PEPn
sym,C⊥= ψ(PEPn,C⊥) = 1 ∣C∣ψ○ ̃M(PEPn,C) = 1 ∣C∣M○ψ(PÊ Pn,C) =∣C∣1M(PÊ Pn sym,C).
Just as for the product partition we may write the identity (6) in the form WEPn
sym,C⊥(Z1,...,ZL) =
1
∣C∣WEPsymn ,C(K(Z1,...,ZL) T),
where again K= (km,l) ∈ CL×Lis the Krawtchouk matrix of the partitionP. For codes
over fields this identity appeared already in MacWilliams and Sloane [18, Ch. 5, Thm. 10].
4
Examples
The MacWilliams identities for symmetrized partitions in Theorem 9 lead to the best known examples. Therefore, we cover these first and start with the most famous one.
Example 10. Consider the partition {0} ∪ (R/{0}) on R. This is indeed an F-
partition as follows immediately from Proposition 3. For the resulting symmetrized partition on Rnthe composition vector as defined in Definition 7(b) is comp(v) = (n− wt(v),wt(v)), where wt(v) denotes the Hamming weight of v ∈ Rn. Hence the induced symmetrized partition on Rnis simply(Ql)l=0,...,n, where Ql= {v ∈ Rn∣ wt(v) = l},
and the symmetrized partition enumerator of a codeC in Rnis the classical Hamming weight enumeratorhweC= ∑v∈CZ0n−wt(v)Z1wt(v). Proposition 3 along with Theorem 9 show that the MacWilliams transform amounts to Z0↦ Z0+(∣R∣−1)Z1, Z1↦ Z0−Z1,
and thus we have the familiar identity hweC⊥(Z0,Z1) =
1
Festschrift in Honor of Uwe Helmke H. Gluesing-Luerssen For fields, this identity is the classical result of MacWilliams [17]. For codes over the
ringZ4it has been derived by Hammons et al. [7, p. 303] (see also the references
therein), for arbitrary residue ringsZmby Klemm [14], for arbitrary finite Frobenius
rings by Nechaev and Kuzmin [20], and finally for non-commutative finite Frobenius rings by Wood [26, Thm. 8.3].
For the following examples, recall from Proposition 5 that each subgroup U⊆ R∗
gives rise to an F-partition.
Example 11. Let U= {1} be the trivial group. Then U induces the partition P
consisting of the singletons{a}, a ∈ R. The resulting symmetrized partition enu-
merator is the complete weight enumerator cweC∶= ∑v∈C∏nt=1Zvt ∈ C[Zα∣ α ∈ R].
The coefficient of∏α∈RZsα
α is the number of codewords having exactly sα entries
equal to α. The MacWilliams identity is given by cweC⊥(Z) = 1
∣C∣cweC(KcZ), where
Z= (zα∣ α ∈ R)Tand Kc= (χ(αβ))α,β∈R. For codes over fields, the identity appears
already in the textbook [18] by MacWilliams and Sloane. Consider the following two special cases.
1) Let R= F4= {0,1,a,a+1}, where a2= a+1. A generating character of F4is given
by χ(0) = χ(1) = 1 and χ(a) = χ(a2) = −1. Thus, Kc= ⎡⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎣ 1 1 1 1 1 1 −1 −1 1 −1 −1 1 1 −1 1 −1 ⎤⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎦ ,
and the MacWilliams identity for the complete weight enumerator reads as
cweC⊥(Z0,Z1,Za,Z
a2) =
1
∣C∣cweC(Z0+Z1+Za+Za2, Z0+Z1−Za−Za2,
Z0−Z1−Za+Za2, Z0−Z1+Za−Za2). (7)
2) For R= Z4a generating character is given by χ(a) = iafor a= 0,...,3, and the
MacWilliams identity is
cweC⊥(Z0,Z1,Z2,Z3) =
1
∣C∣cweC(Z0+Z1+Z2+Z3, Z0+iZ1−Z2−iZ3,
Z0−Z1+Z2−Z3, Z0−iZ1−Z2+iZ3). (8)
It appeared in [7, p. 303] as well as a special case of [14, Satz 1.2].
Example 12. Let R= Zmfor some m∈ N. Put U = {1, −1}. Then the orbits of the
action of U on R are given by P0= {0} and Pa= {a,−a} (which may be a singleton).
By Proposition 5 the orbits form an F-partitionP. It consists of L ∶= ⌊m/2⌋ nonzero
sets. The induced symmetrized partition enumerator on Rnis called the symmetrized
Lee weight enumerator; thus slweC∶= WEPn
sym,C= ∑v∈C∏
n
t=1Z[vt], where, as usual,
[vt] is the index of the partition set {vt,−vt}. It enumerates the codewords having
MacWilliams identity.
Consider for example, the ring R= Z4. We may choose again χ(a) = ia, a= 0,...,3.
In this case we have the partition sets P0= {0}, P1= {1,3}, P2= {2} and obtain the
transform ̃M(Zl) = Z0+(χ(l)+ χ(−l))Z1+ χ(2l)Z2for l= 0,1,2. This results in
the identity
slweC⊥(Z0,Z1,Z2) =
1
∣C∣slweC(Z0+2Z1+Z2, Z0−Z2,Z0−2Z1+Z2), (9)
as it has been presented already in [7, p. 303] as well as in [12, Satz 1.2] as a special case.
Example 13. (1) This example has been studied by Klemm [12]. It generalizes (9) in a particular way. Consider R= Zmand let U= Z∗m. Then the orbits of U in R are
Pd∶= {a ∈ Zm∣ gcd(a,m) = d} for all divisors d of m. Note that P1= U and Pm= {0}.
Hence the coefficient of∏d∣mZsd
d in the symmetrized partition enumerator equals
the number of codewords having exactly sdentries with additive order md−1. The
MacWilliams identity in Theorem 9 tells us that this information about the code fully determines the same information of the dual code.
(2) This example appeared in [10] by Huber. Let R= Fq= Fpm be a field of odd
characteristic p and such that m is even if p≡ 3 mod 4. Then (q−1)/4 ∈ Z and thus
there exists an element i inFqsuch that i2= −1. Define U ∶= ⟨i⟩ = {1,−1,i,−i}. Its
orbits inFqform an F-partitionP consisting of L ∶= (q−1)/4+1 sets. For a vector
v∈ Fnq, the composition vector compP(v) ∈ NL0counts the number of entries of v in each orbit, see Definition 7(b). It is called the Gaussian weight in [10]. Theorem 9 provides a MacWilliams identity for the resulting partition enumerator, which has already been presented in [10, Thm. 2].
Let us now turn to examples for the MacWilliams identity for product partition enumerators as derived in Theorem 8. We obtain the well-known identity for the exact weight enumerator as well as some other, lesser known, identities.
Example 14. This is the de-symmetrized version of the complete weight enumer-
ator discussed in Example 11. Let U be the trivial group{1}, which induces the
partitionP consisting of the singletons {a}. The resulting product partition enu-
merator PEPn,Cis the exact weight enumerator eweC∶= ∑v∈C∏nt=1Yt,v
t ∈ C[Yt,α∣ t =
1,...,n, α∈ R]. The monomials of this polynomial are in bijection to the codewords. Just like the full weight enumerator in Theorem 6, the exact weight enumerator carries all information about the code (this time, the information is encoded in a
polynomial in n∣R∣ indeterminates, whereas the full weight enumerator is a poly-
nomial in∣R∣nindeterminates). It is thus clear that the exact weight enumerators
of the code and its dual must determine each other, and the MacWilliams identity from Theorem 8 simply makes this explicit. The associated MacWilliams transform is given by ̃M(Yt,α) = ∑β∈Rχ(αβ)Yt,β for all t. For codes over fields this identity
appears already in [18, Ch. 5, Thm. 14] by MacWilliams and Sloane. The next examples cover in particular the Lee weight.
Festschrift in Honor of Uwe Helmke H. Gluesing-Luerssen Example 15. 1) This is the de-symmetrized version of Example 12. Let R, U and the partitionP be as in Example 12. The product partition enumerator is based on the weight function given by v↦ ∏nt=1Zt,[vt]∈ C[Zt,l∣ t = 1,...,n, l = 0,...,L], where L is the number of nonzero U -orbits. The enumerator keeps track of the entries of v up to sign, but including their position t.
As a special case, let R= Zmand χ∶ Zm→ C, a ↦ ζa, where ζ∈ C is a primitive m-th
root of unity. We call the resulting product partition enumerator plweCthe Product Lee Weight Enumerator. The coefficient of a monomial∏tn=1Zt,lt equals the number
of all codewords for which the t-th entry is±lt. According to Theorem 8, plweCand
plweC⊥satisfy a MacWilliams identity with MacWilliams transform given by
̃
M(Zt,l) =⎧⎪⎪⎨⎪⎪
⎩
Zt,0+∑Ls=1(ζl⋅s+ζ−l⋅s)Zt,s if m is odd
Zt,0+∑Ls=1−1(ζl⋅s+ζ−l⋅s)Zt,s+ζl⋅LZt,L if m is even
2) In the same way there exists a de-symmetrized version of Example 13(1). The analogous product partition enumerator keeps track of the additive order of each individual codeword coordinate. Again the MacWilliams identity tells us that this information fully determines the same type of information for the dual code. The following is the de-symmetrized version of the Hamming weight.
Example 16. Consider again the F-partition{0}∪(R/{0}) on R as in Example 10.
The induced product partition on Rn leads to the weight function given by v↦
∏n
t=1St,[vt], where[0] = 0 and [a] = 1 for a ≠ 0. As a consequence, the product parti-
tion enumerator is the Support Tracker supp-weC∶= ∑v∈C∏nt=1St,[vt]. The coefficient of the monomial∏nt=1St,lt enumerates the codewords having support equal to the set {t ∣ lt= 1} ⊆ {1,...,n}. The MacWilliams identity obtained from Theorem 8 has been
discussed already by Zinoviev and Ericson [28, Ex. 3] and Honold and Landjev [9, Ex. 23].
Example 17. This example does not immediately fit into our setting, but can be
dealt with via a simple adjustment. LetF be a finite field. For a nonzero vector
v= (v1,...,vn) ∈ Fndefine ρ(v) ∶= max{i ∣ vi≠ 0} and put ρ(0) = 0. Then ρ induces
a metric onFn, the Rosenbloom-Tsfasman weight, which has been introduced by
Rosenbloom and Tsfasman in [21]. This metric plays a specific role for matrices, to which it generalizes straightforwardly by taking the sum of ρ(x) over all rows x of the matrix (see also [3]). For the relevance of the Rosenbloom-Tsfasman weight for detecting matrix codes with large Hamming distance see [23].
We will now derive a MacWilliams identity for the Rosenbloom-Tsfasman enumerator of codes inFnand their reversed dual. For i= 0,...,n let P
i= {v ∈ Fn∣ ρ(v) = i}, and
letP be the partition (Pi)ni=0ofFn. Note thatP is not induced by a partition on F as
in Definition 7. For a codeC ⊆ Fndefine the corresponding Rosenbloom-Tsfasman
enumerator RTC= ∑ni=0∣C ∩ Pi∣Zi. Then RTC∈ C[Z0,...,Zn]hom,1, the space of all
homogeneous polynomials of degree 1 in n+1 variables.
Examples show immediately that this enumerator does not satisfy a MacWilliams identity for a codeC and its dual C⊥. However, if one redefines the dual by applying
a coordinate reversal, then a MacWilliams identity can be derived. Define J∶= ⎡⎢ ⎢⎢ ⎢⎢ ⎢⎣ 1 ... 1 ⎤⎥ ⎥⎥ ⎥⎥ ⎥⎦∈ F n×n. (10)
Then J= J−1= JT. The partitionP has the following invariance property. First of all, it is not hard to see that the sets Piform the orbits of the group action of the invertible
lower triangular matrices onFn. Next, let v,v′∈ Pifor some i, thus v′= vA for some
invertible lower triangular matrix A. Notice that JATJis again lower triangular.
Hence, Pj(JATJ) = Pjfor each partition set Pj, and one easily derives
∑
w∈Pj
χ(⟨w,v′J⟩) = ∑
w∈Pj
χ(⟨w,vJ⟩).
As a consequence, the (generalized) Krawtchouk coefficients ki j= ∑w∈Pjχ(⟨w,vJ⟩),
where v∈ Pi, depend only on i, j, and not on the specific choice of v∈ Pi.
For a codeC ⊆ Fndefine the reversed dual codeC⍊∶= C⊥J∶= {wJ ∣ w ∈ C⊥}. Now it is
easy to derive a MacWilliams identity between RTCand RTC⍊. Consider again the
full weight enumerator from Theorem 6, and for v∈ Fnlet τ(Xv) = Zρ(v). As usual,
we extend τ to an algebra homomorphism onC[Xv∣ v ∈ Fn]. Using the MacWilliams
transformM from Theorem 6 one obtains
τ○M(XvJ) = ∑ w∈Fn χ(⟨w,vJ⟩)Zρ(w)= n ∑ j=0w∑∈Pj χ(⟨w,vJ⟩)Zj= M′○τ(Xv),
where the transformM′onC[Z0,...,Zn]hom,1is defined as the vector space isomor-
phism given by M′(Zi) = ∑nj=0ki jZj. Note that this is a linear map. Along with
Theorem 6 all of this now leads to the MacWilliams Identity RTC⍊= τ(fweC⊥J) = 1 ∣C∣τ○M(fwe(C⊥J)⊥) = 1 ∣C∣τ○M(fweCJ) =∣C∣M1 ′○τ(fwe C) =∣C∣M1 ′(RTC).
The Krawtchouk coefficients for the transform can be computed explicitly. Consider- ing⟨w,vJ⟩ and making use of the properties in Proposition 3, one derives
ki, j= ⎧⎪⎪⎪⎪ ⎪⎪⎪ ⎨⎪⎪⎪ ⎪⎪⎪⎪ ⎩ 1 if j= 0 ∣Pj∣ = qj−1(q−1) if 1 ≤ j ≤ n−i −qn−i if j= n−i+1 0 if j> n−i+1
Thus the MacWilliams transform reads as M′(Zi) = Z0+
n−i
∑
j=1
qj−1(q−1)Zj−qn−iZn−i+1. (11)
This has also been derived in [3, Thm. 3.1]. We come back to this in Example 21, when it will be extended to the space of matricesFs×n.
Festschrift in Honor of Uwe Helmke H. Gluesing-Luerssen