Partitions of Frobenius Rings Induced by the Homogeneous Weight

The values of the homogeneous weight are determined for finite Frobenius rings that are a direct product of local Frobenius rings. This is used to investigate the partition induced by this weight and its dual partition under character-theoretic dualization. A characterization is given of those rings for which the induced partition is reflexive or even self-dual.


Introduction
The homogeneous weight has been studied extensively in the literature of codes over rings. It has been introduced by Constantinescu and Heise [6] as a generalization of both the Hamming weight on finite fields and the Lee weight on Z 4 . Its main feature is that the average weight of the elements in a nonzero principal ideal is the same constant for all such ideals. The weight has been further generalized to arbitrary non-commutative finite rings by Greferath and Schmidt [16] as well as Honold and Nechaev [20]. The homogeneous weight has proven to be an important tool in ring-linear coding. For instance, in [10] Duursma et al. construct non-linear codes with the best parameters so far using certain ring-linear codes and where the ring is endowed with the homogenous weight.
These and other properties of the homogeneous weight have led to a detailed study of this weight. Among other things, it has been shown that the MacWilliams extension theorem remains true for isomorphisms preserving the homogeneous weight, see Constantinescu et al. [7] for codes over the integer residue ring Z N , Wood [25] and Greferath and Schmidt [16] for codes over general finite Frobenius rings, and Greferath et al. [14] for codes over the Frobenius module of a finite ring.
On the other hand, so far no explicit MacWilliams identity for the homogeneous-weight enumerators of codes has been established in any general form. Such identities relate a suitably defined weight enumerator of a code to the weight enumerator (or a dual version thereof) of its dual code. MacWilliams identities are well known for many weight functions, e.g., the Hamming weight, the complete weight, the symmetrized Lee weight, and more.
The non-existence of an explicit MacWilliams identity for the homogeneous weight is due to the fact that the partition induced by this weight does not behave as well under dualization as those for the Hamming weight or the other weights just mentioned. More precisely, the induced partition is in general not self-dual with respect to a certain character-theoretic dualization. In this paper we will study the homogeneous weight partition on a certain class of finite Frobenius rings, which includes all commutative Frobenius rings, and will provide a characterization of those rings, for which the partition is reflexive (that is, coincides with its bidual) or even self-dual. Reflexivity, which is weaker than self-duality, guarantees a MacWilliams identity because in this case the partition and its dual have the same number of partition sets. In this case we will also provide the associated Krawtchouk coefficients. With these data, an explicit MacWilliams identity relating the corresponding partition enumerators is simply an instance of the general theory about reflexive partitions, see for instance [13] or Camion [4]. A precursor of these ideas is the paper [3] by Greferath et al. We will study the homogeneous weight on Frobenius rings that are a direct product of local Frobenius rings. For the latter the homogeneous weight is well-known [3] and takes a very simple form. Making use of an explicit formula for the values of the homogeneous weight provided by Honold [18], we will be able to compute the values of the homogeneous weight on the specified Frobenius rings. This is carried out in Section 3.
In Section 4 we then go on and study the partition induced by the homogeneous weight. We will see that all elements outside the socle have the same weight, thus form one partition set, whereas in the socle the homogeneous partition is closely related to the product of Hamming partitions that are induced by a suitable direct product representation of the ring. In fact, we will show that the homogeneous partition is reflexive if and only if its restriction to the socle coincides with the just described product of the Hamming partitions. We also give a characterization of the rings for which the homogeneous partition is reflexive. It is given in terms of the orders of the residue fields of the local component rings. Finally, we will prove that the homogeneous weight partition is self-dual if and only if it is reflexive and the ring is semisimple.

Frobenius Rings and Partitions
Throughout this section, let G = (G, +) be a finite abelian group, and let R be a finite ring with unity. Denote its group of units by R * . Our main subject is partitions and their duals of R or R n , but occasionally we need to consider partitions of the group; mainly for the situation where G is the additive group of an ideal of R. For this reason we present the main notions of this section for groups with special emphasis on rings.
Denote by G the complex character group of G. Thus G = Hom G, C * is the set of all group homomorphisms from G into C * with addition (χ 1 + χ 2 )(a) := χ 1 (a)χ 2 (a). The zero element is the principal character ε ∈ G, given by ε(a) = 1 for all a ∈ G. It is well known that G and G are isomorphic. The most fundamental property of characters on G is the orthogonality relation For a ring R and n ∈ N we denote by R n the character group of the additive group (R n , +). This group can be endowed with an R-R-bimodule structure via the left and right scalar multiplications (r·χ)(v) = χ(vr) and (χ·r)(v) = χ(rv) for all r ∈ R and v ∈ R n . (2.2) Recall that R is a Frobenius ring if R soc( R R) ∼ = R (R/rad(R)), where soc( R R) denotes the socle of the left R-module R and rad(R) is the Jacobson radical of R (it is well-known that the existence of a left isomorphism implies the right analogue). Since rad(R) is a two-sided ideal, R/rad(R) is even a ring. Furthermore, soc( R R) = soc(R R ), and we will simply write soc(R) for the socle.
We summarize the following properties about Frobenius rings, some of which actually characterize the Frobenius property, but we will not engage in that discussion. Details can be found in many books on ring theory, e.g., [22,Ch. 6] by Lam or in the research articles by Lamprecht [23], Hirano [17], Wood [26,Thm. 3.10], and Honold [18].
Remark 2.1. Let R be a finite Frobenius ring. Then the following are true. (a) R andR are isomorphic left R-modules and isomorphic right R-modules. More precisely, there exists a character χ ∈R such that is an isomorphism of left (resp. right) R-modules. Any such χ is called a generating character of R. Obviously, any two generating characters χ, χ ′ differ by a unit, i.e., χ ′ = u·χ and χ ′ = χ·u ′ for some u, u ′ ∈ R * . More generally, for each n, the maps are left (resp. right) R-module isomorphisms (here v, w denotes the standard inner product on R n ). (b) Let χ be a character of R. Then χ is a generating character of R if and only if the only left (resp. right) ideal contained in ker χ := {a ∈ R | χ(a) = 1} is the zero ideal; see [5,Cor. 3.6]. (c) R satisfies the double annihilator property, i.e., ann l (ann r (I)) = I for each left ideal I of R and ann r (ann l (I)) = I for each right ideal I, where ann l (resp. ann r ) denotes the left (resp. right) annihilator ideal; see [22,Thm. 15.1]. In particular, ann l (rad(R)) = soc(R) = ann r (rad(R)), see [22,Cor. 15.7]. (d) soc(R) is a principal ideal [16, p. 21]. (e) If R is a commutative finite Frobenius ring, then R = R 1 × . . . × R t for suitable local Frobenius rings R i ; see [22,Th. 15.27]. (f) If R is local, that is, rad(R) is the unique maximal left (resp. right) ideal of R, then soc(R) is the unique minimal ideal [22,Ex. (3.14)]. In this case R/rad(R) is called the residue field of R. character is of the form χ(g) := ζ g for all g ∈ Z N , where ζ ∈ C is an N -th primitive root of unity. Thus, each character is given by (a·χ)(g) = χ(ag) = ζ ag for some a ∈ Z N . (b) Every finite field is Frobenius, and every non-principal character is generating. (c) Finite chain rings (e.g., Galois rings), finite group rings over a Frobenius ring and direct products of Frobenius rings are Frobenius. For a subgroup H ≤ G (called an additive code) we define the dual subgroup as (2.5) It is straightforward to see that As usual, a code over R is defined to be a left submodule of R n for some n ∈ N. Following the above, the dual code is C ⊥ = {χ ∈ R n | χ(v) = 1 for all v ∈ C}. It is easily seen that for any code C the dual C ⊥ is a right R-submodule of R. The double annihilator property from Remark 2.1(c) extends to C ⊥⊥ = C and |C ⊥ ||C| = |R n |, see [19,Cor. 5] or [13,Rem. 5.5]. Applying the isomorphisms in (2.4) to the additive group of C ⊥ results in the right and left hand side dual (see also [26,Thm. 7.7]) (2.7) Note that C ⊥,l is an R-R-bimodule, whereas C ⊥,r is in general just a right R-module.
We now turn to partitions on G and fix the following notation. A partition P = (P m ) M m=1 of a set X will mostly be written as P = P 1 | P 2 | . . . | P M . The sets of the partition are called its blocks, and we write |P| for the number of blocks in P. Recall that two partitions P and Q are called identical if |P| = |Q| and the blocks coincide after suitable indexing. Moreover, P is called finer than Q (or Q is coarser than P), written as P ≤ Q, if every block of P is contained in a block of Q. Note that if P ≤ Q then |P| ≥ |Q|. Denote by ∼ P the equivalence relation induced by P, thus, v∼ P v ′ if v, v ′ are in the same block of P.
The following notion of a dual partition will be crucial for us. The left-sided version has been introduced for Frobenius rings by Byrne et al. [3, p. 291] and goes back to the notion of F-partitions as introduced by Zinoviev and Ericson in [27]. Reflexive partitions, defined below, appear already in [28] by Zinoviev and Ericson were they have been coined B-partitions. They are exactly the partitions that induce abelian association schemes as studied in a more general context by Delsarte [8], Camion [4], and others, see also [9]. For an overview of these various approaches and their relations in the language of partitions, see also [13].
Throughout, we will use the notation   , (2.11) where P is the bidual partition in the group sense of (2.8 , which is equivalent to P = . Moreover, Let us briefly discuss the Krawtchouk coefficients K ℓ,m of the pair (P, P) from Definition 2.3(a). Due to the very definition of the dual partition, these coefficients do not depend on the choice of χ in Q ℓ . In the ring setting the coefficients read as follows. Let In particular, K ℓ,m does not depend on the choice of v inside Q ′ ℓ (resp. Q ′′ ℓ ), and neither does it depend on the sidedness of the dual partition.
The following example illustrates that the dual partition does in general depend on the choice of the generating character χ, even in the commutative case.
Remark 2.6. For the integer residue rings R = Z N , the dual of a partition does not depend on the choice of the generating character. This follows immediately from Example 2.2(a) along with the fact that all primitive N -th roots of unity have the same minimal polynomial. The latter implies that the identities on the right hand side of (2.9) do not depend on the choice of the primitive N -th root ζ.
For many standard partitions on Frobenius rings, e.g., the Hamming partition, the dual does not depend on the choice of the generating character. In the next section we will see that this is also the case for the main topic of this paper, partitions induced by the homogeneous weight.
Let us return to the general situation of partitions of groups. We have the following simple observation.
Remark 2.7. (a) The singleton {ε} is always a block of P. Indeed, g∈Pm ε(g) = |P m | for every block P m . Hence if χ ∈ G satisfies g∈Pm χ(g) = |P m | for all P m , then g∈G χ(g) = |G| and thus χ = ε by (2.1). Using (2.4) we also conclude that {0} is a block of P [χ,l] and P [χ,r] . r] . This follows directly from the definition of the dual partitions since each block of Q is the union of blocks of P.
As has been shown in various forms in the literature [8,28,3,13], a partition P of R n and its dual partition P of R n allow a MacWilliams identity: applying a certain MacWilliams transformation to the P-partition enumerator of a code C ⊆ R n results in the P-enumerator of its dual C ⊥ ⊆ R n . For an overview in the language of this paper see [13,Sec. 2]. The most symmetric situation arises for reflexive (or even self-dual) partitions, in which case the transformation can be carried out in both directions, and thus the two enumerators determine each other uniquely. Most, if not all, classical examples of MacWilliams identities are instances of this general MacWilliams identity based on a self-dual partition (for instance, for the Hamming weight, symmetrized Lee weight, complete weight, and the rank metric).
In the next section we will study the partition on a Frobenius ring induced by the homogeneous weight and investigate for which rings the partition is reflexive or even self-dual. Our main tool for characterizing reflexivity of a partition is the following convenient criterion from [13, Thm. 3.1].
Theorem 2.8. For any partition P on G and its dual partition P we have |P| ≤ | P| and P ≤ P. Moreover, P is reflexive if and only if |P| = | P|.
We close this section with several specific instances of reflexive partitions that will be needed in the next section.
Example 2.9. Let R be a Frobenius ring with generating character χ, and denote by P * ,l (resp. P * ,r ) the partition given by the orbits of the left (resp. right) action of R * on R. Thus, the blocks of P * ,l are given by the distinct orbits O x,l = {ux | u ∈ R * }, x ∈ R, whereas the blocks of P * ,r are given by the orbits = P * ,r and P * ,r [χ,l] = P * ,l . In particular, the dual partitions do not depend on the choice of χ, and the partitions are reflexive.
The Hamming partition and its Krawtchouk coefficients will be needed in the following form in the next section.
, the characters of G are given by Assume |A i | = q for all i, and let P be the partition of G induced by the Hamming weight, that is, . . , n}, and where wt(a 1 , . . . , a n ) = |{i | a i = 0}|. Then it is well known that the dual partition P is the partition induced by the Hamming weight on A 1 × . . . × A n . In other words, the blocks of P are is the Krawtchouk polynomial (see for instance [8,Thm. 4.1] or Lemma 2.6.2 in [21]; the proof of Lemma 2.6.2 in [21], given for Z n q , works mutatis mutandis for all A 1 × . . . × A n ). All of this shows that the Hamming partition on G is reflexive. Finally, the Hamming partition on the module R n is self-dual with respect to any generating character χ by virtue of (2.4).
When studying the partition induced by the homogeneous weight, we will need to consider product partitions. The following result will suffice for us. Let G = A 1 × · · · × A n , where A 1 , . . . , A n are finite abelian groups and let P i be reflexive partitions of A i for i = 1, . . . , n. Write P i = P i,1 | P i,2 | . . . | P i,M i . Then the product partition on G is defined as . The dual partition of Q is Q = P 1 × · · · × P n , and in particular, Q is reflexive. Furthermore, the Krawtchouk coefficients of (Q, Q) are are the Krawtchouk coefficients of (P i , P i ). The following result shows that the trivial extension of a partition of a subgroup behaves well under dualization. The notation χ |H stands for the restriction of χ to H, whereas ε G and ε H denote the principal characters on G and H, respectively.
Proof. We have to consider various cases. Evidently, in all of these cases the sums do not depend on the specific choice of χ within the specified set, and thus the partition Q : . . | Q ′ L is finer than or equal to P ′ . The above also establishes the Krawtchouk coefficients stated in (2.16). Since P is the dual partition of P, Definition 2.3(a) implies that no two rows of the matrix in (2.16) coincide. This means that if χ ∈ Q ′ ℓ and χ ′ ∈ Q ′ ℓ ′ , where ℓ = ℓ ′ , then χ ∼ P ′ χ ′ . Thus P ′ = Q, as desired. The statement concerning reflexivity follows from Theorem 2.8.

Explicit Values of the Homogeneous Weight
In this section we consider the homogeneous weight and determine its values for those finite Frobenius rings that are isomorphic to a product of local rings. Due to Remark 2.1(e) this includes all finite commutative Frobenius rings. In the subsequent section, the results will be used to study the partition induced by the homogeneous weight.
Throughout, let R be a finite Frobenius ring with group of units R * , and fix a generating character χ. The following definition is taken from Greferath and Schmidt [16].
which will be useful later for determining the values of the homogeneous weight on arbitrary Frobenius rings.
Definition 3.2. The partition of R induced by homogeneous weight is denoted by P hom . It is thus given by the equivalence relation x∼ P hom x ′ ⇐⇒ ω(x) = ω(x ′ ) for x, x ′ ∈ R. In other words, the blocks of P hom are the sets consisting of all elements sharing the same homogeneous weight.
Recall the partitions P * ,l and P * ,r induced by the left and right action of R * on R, see Example 2.9. Definition 3.1(ii) implies P * ,l ≤ P hom and (3.1) yields P * ,r ≤ P hom . Thus, P * ,r ≤ P hom [χ,r] and P * ,l ≤ P hom [χ,l] due to Example 2.9 and Remark 2.7(b).
Using the fact that the orbit O x,r of x ∈ R is given by O x,r = {xu 1 , . . . , xu m }, where u 1 , . . . , u m are representatives of the distinct right cosets of the stabilizer subgroup of x ∈ R, one obtains b∈Ox,r Now we can summarize the following simple properties, which will be needed later when computing the dual partition of P hom . The latter will be defined as the partition P hom [χ,r] of R in the sense of Definition 2.3(b). We will now see that the dual does not depend on the choice of χ.
This shows that the sum b∈P χ(ab) does not depend on the choice of the generating character χ.
As a consequence, the dual partition P hom [χ,r] does not depend on χ. We will therefore simply denote this partition by P hom r . Thus a∼ P hom r a ′ ⇐⇒ b∈P χ(ab) = b∈P χ(a ′ b) for all blocks P of P hom .
Note also that the Krawtchouk coefficients of (P hom , P hom r ) are real. One may also observe that the above implies P * ,l ≤ P hom r . Similar statements can be obtained for the left dual of P hom , which then also does not depend on χ.
The following examples illustrate that the homogeneous partition on a Frobenius ring may display a variety of different properties. Since all examples are commutative we will simply write P hom instead of P hom r . Example 3.4. (a) On Z 8 the homogeneous partition is given by P hom = 0 | 1, 2, 3, 5, 6, 7 | 4. Its dual is P hom = 0 | 1, 3, 5, 7 | 2, 4, 6, which has been observed already in [3, Ex. 2.9]. Thus P hom is not self-dual, but reflexive, due to Theorem 2.8. (b) On R = Z 2 × Z 2 the homogeneous partition is easily seen to be 00, 11 | 01, 10, and the values of the weight are 0 and 2, respectively. The fact that {00} is not a block of P hom shows that P hom is not the dual of any partition; see Remark 2.7(a). In particular, P hom is not reflexive and thus not self-dual. In Remark 3.10(b) a characterization will be presented for the Frobenius rings that contain a nonzero element with homogeneous weight zero. (c) Consider R = Z 3 ×Z 3 . In this case one easily verifies that P hom = 00 | 10, 20, 01, 02 | 11, 12, 21, 22 with normalized homogeneous weights 0, 3/2, and 3/4, respectively. Note that P hom is simply the Hamming partition on Z 3 × Z 3 , and thus reflexive and even self-dual, i.e., P hom = P hom . This will also follow from Theorem 4.4.
In order to determine the values of the homogeneous weight explicitly, we start with the following well-known case.
Example 3.5. Let R be a local Frobenius ring with residue field R/rad(R) of order q. Then the normalized homogeneous weight is given by This can be verified immediately using the fact that soc(R) is the unique minimal left ideal [22,Ex. (3.14)] and thus contained in any nonzero left ideal of R; see also [3,Ex. 2.8] for an argument involving the Möbius function for the lattice of ideals of R. Hence |P hom | = 3. In Theorem 4.4 (see also Example 4.5(a)) we will see that P hom is reflexive, and we will also determine the dual partition.
We now proceed to determine the values of the homogeneous weight for finite Frobenius rings that can be written as the direct product of local Frobenius rings. Thanks to Remark 2.1(e), this covers all finite commutative Frobenius rings. We first summarize some basic properties for direct products of Frobenius rings.
The above leads to the following identity for the values of the homogenous weight. A similar formula, but in terms of the Möbius function and for the case where all R i are commutative principal ideal rings, appeared in [12,Thm. 4.1], see [11, p. 4].
Proposition 3.7. Let R = R 1 ×. . .×R t be a direct product of (not necessarily local) Frobenius rings. For i ∈ [t] let ω i be the normalized homogeneous weight on R i . Then the normalized homogeneous weight on R is given by for (a 1 , . . . , a t ) ∈ R.

Now (3.1) leads to the desired identity.
This result brings us immediately to an explicit formula for the homogeneous weight in the following case.
where each R i is a finite local Frobenius ring with residue field R i /rad(R i ) of order q. Then where wt(a) := |{i | a i = 0}| denotes the Hamming weight of a = (a 1 , . . . , a t ).
Proof. This is a consequence of Example 3.5 along with Proposition 3.7 and the fact that soc(R) = soc(R 1 ) × . . . × soc(R t ).
In the same way we can compute the homogeneous weight on any finite Frobenius ring that is given as as a direct product of local rings. We will need to keep track of the orders of the residue fields of the component rings and thus fix the following notation. For the rest of this paper, let R be a finite Frobenius ring of the form |R i,j /rad(R i,j )| = |soc(R i,j )| = q i for all j ∈ [n i ] and q 1 , . . . , q t distinct. (3.6) Recall that soc(R i,j ) ∼ = R i,j /rad(R i,j ). Propositions 3.7 and 3.8 yield the following generalization of Example 3.5.
Theorem 3.9. Let R be as in (3.6) and write its elements as a = (a 1 , . . . , a t ), where a i ∈ R i . Using the Hamming weight wt on each R i , the homogeneous weight on R is given by We close this section with the following immediate insight about the induced partition. A detailed study of the homogeneous partition and its dual is presented in the next section.

The Partition Induced by the Homogeneous Weight
We characterize the rings for which the partition P hom induced by the homogeneous weight is reflexive or even self-dual. Thanks to Proposition 2.4 we may and will restrict ourselves to the right dual of P hom in order to study self-duality (and reflexivity). In the reflexive case we also determine the right dual partition and the Krawtchouk coefficients explicitly. Recall from Remark 3.3 that the right dual partition of P hom in the sense of Definition 2.3(b) does not depend on the choice of the generating character and is denoted by P hom r . Some basic properties of P hom were presented already in Remark 3.10. We now focus on a simple consequence of Theorem 3.9 that will turn out to be crucial for characterizing reflexivity. Recall that R is as in (3.6). Theorem 3.9 shows that the homogeneous weight of a = (a 1 , . . . , a t ) ∈ R depends on the Hamming weights wt(a i ). In particular, if a, b ∈ soc(R) are such that wt(a i ) = wt(b i ) for all i ∈ [t], then ω(a) = ω(b). This shows that the homogeneous partition is closely related to the product of the Hamming partitions on soc(R i ), i ∈ [t].
We will therefore study this product partition first and come back to the homogeneous partition thereafter. As we will show later, the homogeneous partition is reflexive if and only if it coincides on the socle with the product of the Hamming partitions.
Denote by H i the Hamming partition on soc(R i ) = soc(R i,1 ) × . . . × soc(R i,n i ), thus H i = P i,0 | . . . | P i,n i with blocks P i,j = {(a i,1 , . . . , a i,n i ) ∈ soc(R i ) | wt(a i,1 , . . . , a i,n i ) = j}. The induced product partition on soc(R 1 ) × . . . × soc(R t ) = soc(R) is given by see (2.15), and consists of the blocks For the dual partition the following identifications are useful. Note first that R/rad(R) ∼ = R 1 /rad(R 1 ) × . . . × R t /rad(R t ) for R as in (3.6). In the same way This allows us to consider the Hamming weight on R i /rad(R i ). For a i = (a i,1 , . . . , a i,n i ) ∈ R i put wt a i + rad(R i ) := |{j | a i,j ∈ rad(R i,j )}|. (4.4) Now we can formulate the following duality. In particular, H ′ is reflexive. If R = soc(R) then |H ′ | = s := t i=1 (n i + 1), and H ′ is simply the product of the Hamming partitions and thus self-dual. If R = soc(R) then |H ′ | = s + 1.
Proof. We make use of Theorem 2.11 and Proposition 2.12. For this we consider soc(R) as a subgroup of R. Then H is a partition of this subgroup, and it is given as the product of the Hamming partitions H i of soc(R i ). By Example 2.10, the dual partitions H i are the Hamming partitions on the character groups soc(R i ), and Theorem 2.11 implies that H is the partition H 1 × . . . × H t of soc(R) = soc(R 1 ) × . . . × soc(R t ). Proposition 2.12 yields that the partition H ′ of the group R consists of the blocks soc(R) ⊥ \{ε}, {ε}, and the blocks for all m ∈ M\{0}, and where χ i is a fixed generating character of R i (see also Remark 3.6). Now the isomorphism α r from (2.4) turns the partition H ′ of R into the partition H ′ [χ,r] of R. Since α −1 r (soc(R) ⊥ ) = ann l (soc(R)) = rad(R) due to (2.7), it is clear that the partition H ′ [χ,r] consists of the blocks rad(R)\{0}, {0}, and the sets Q m given in the theorem. All of this proves the desired duality. Evidently, the left-sided version of (4.5) is true as well, and the analogous proof establishes H ′ [χ,r] = H ′ [χ,l] . The cardinality of H ′ is clear from |M| = s, and reflexivity follows from Theorem 2.8. Finally, if R = soc(R), then rad(R) = {0}, so that in this case the blocks P ⋄ and Q ⋄ are missing, and the sets Q m are the blocks of the product partition H on soc(R) = R. Therefore H ′ is self-dual.
From now on we will simply write H ′ for the dual partition.
for all ℓ, m ∈ M ∪ {⋄}, and where K (n i ,q i ) m i (ℓ i ) are the Krawtchouk coefficients of the Hamming partition of soc(R i,1 ) × . . . × soc(R i,n i ). In the special case where R = soc(R), and thus each R i,j is a field, only the last case occurs.
Proof. This is a consequence of Proposition 2.12 along with Theorem 2.11 and the classical Krawtchouk coefficients for the Hamming partition given in Example 2.10. Now we can return to the homogeneous weight. Note first that the equivalence relation corresponding to the partition H ′ is given by A comparison to the homogeneous weight in Theorem 3.9 shows that H ′ ≤ P hom . In other words, The converse of this implication is not true in general.
In other words, P hom may be strictly coarser than H ′ . Theorem 3.9 indicates that the particular values of q 1 , . . . , q t decide on the difference between these two partitions. We cast the following definition. It is simply made to reflect the case where H ′ = P hom , as we will show in Theorems 4.4 and 4.7.
We call a list [q 1 , . . . , q t ] of distinct prime powers separating, if [(q 1 , 1), . . . , (q t , 1)] is separating. An integer N is separating if its list of distinct prime factors is separating.
As we will see below, for a separating list L the partition H ′ separates the elements of R according to their homogeneous weight.
Starting with three distinct primes and using sufficiently large primes one easily shows that there exist infinitely many separating integers.
Theorem 4.4. Let R be as in (3.6). Suppose the list L = [(q 1 , n 1 ), . . . , (q t , n t )] is separating. Then P hom = H ′ , where H ′ is as in Theorem 4.1. As a consequence, P hom is reflexive and P hom l = P hom r = H ′ . Moreover, if R = soc(R), i.e., R is semisimple, then the homogeneous partition coincides with the product of the Hamming partitions H i on each R i , and thus is self-dual.
Proof. First, the separating property guarantees that if q i = 2 then n i = 1. With Remark 3.10(b) we conclude that ω(a) = 0 for all a = 0. Thus {0} is a block of P hom . Moreover, R \ soc(R) is a block of P hom , see Remark 3.10(a). Since both sets are also blocks of H ′ and H ′ ≤ P hom , it remains to show that for all a, b ∈ soc(R) such that ω(a) = ω(b) we have a∼ H ′ b. By Theorem 3.9 ω(a) = ω(b) yields wt(a i ) and similarly for b. Since q i − 1 > 0, this leads to wt(a) ≡ wt(b) mod 2 and t i=1 (q i − 1) wt(a i ) = t i=1 (q i − 1) wt(b i ) . Since L is separating, this yields wt(a i ) = wt(b i ) for all i ∈ [t], and therefore a∼ H ′ b. This concludes the proof.
Before turning to the non-separating case, let us present some examples of the homogeneous weight on integer residue rings. We use the notation from Theorem 4.1. The last result allows us to simply write P hom for the dual partitions. The Krawtchouk matrix, indexed row-and columnwise by the partition sets of P hom and P hom in the given order, is  For R = Z 8 , this matrix also appears in [4, p. 1553] by Camion.
(b) Let R = Z p r × Z q s , where p, q are distinct primes and r, s ≥ 1. The list [p, q] is separating. Note that R ∼ = Z N , where N = p r q s , and hence N is separating. For the component rings, socle and radical satisfy soc(Z p r )\{0} = O p r−1 (the (Z p r ) * -orbit) and rad(Z p r ) = (p) and analogously for Z q s . The homogeneous partition P hom and the values of the homogeneous weight are given by (in the order P (0,0) | P (1,0) | P (0,1) | P (1,1) | P ⋄ ) The dual partition is P hom = 0 (p)×(q)\{0} (p)×Z * q s Z * p r ×(q) Z * p r ×Z * q s , and the Krawtchouk matrix has the form If N = pq, the last block of P hom and the second block of P hom are missing, and so are the last column and second row of K. For this case, the values of the homogeneous weight also appears in [2, Ex. 3] by Byrne.
Here is the smallest non-separating integer N . As one may expect, the homogeneous partition is not reflexive.
The situation of the last example is true for all non-separating cases.
Theorem 4.7. Let R be as in (3.6) and assume that L = [(q 1 , n 1 ), . . . , (q t , n t )] is non-separating. Then the partition P hom is not reflexive. More precisely, P hom is strictly coarser than the partition H ′ from Theorem 4.1 whereas P hom r = H ′ . Thus | P hom r | > |P hom |. shows that for m ∈ M From the fact that L is non-separating it follows that there exist two distinct partition sets P m , P m ′ for some m, m ′ ∈ M such that all elements of P m ∪ P m ′ have the same homogeneous weight. Thus P hom > H ′ .
We have to show that P hom r = H ′ . In order to do so, we need some preparation. Evidently, any partition set P of P hom , where P = P ⋄ , is of the form P = ∪ m∈L P m for a subset L ⊆ M. Then b∈P χ(ab) = m∈L b∈Pm χ(ab) = m∈L K ℓ,m for any a ∈ Q ℓ , (4.8) where H ′ = (Q ℓ ) ℓ∈M∪{⋄} ; see (2.13).
Let ℓ, ℓ ′ ∈ M ∪ {⋄} be such that Q ℓ and Q ℓ ′ are contained in the same partition set of P hom r .
We have to show that ℓ = ℓ ′ . Let us first convince ourselves that we may assume that ℓ, ℓ ′ are in M\{0} for otherwise we are done. To do so, note that Q 0 = {0} is a partition set of P hom r as this is a general property of dual partitions, see Remark 2.7(a). Thus ℓ = 0 = ℓ ′ . Next, recall that the index ⋄ occurs only if R = soc(R) and that in this case P ⋄ is a block of P hom . Corollary 4.2 shows that K ℓ,⋄ = 0 for all ℓ ∈ {0, ⋄} whereas K ⋄,⋄ < 0 and K 0,⋄ > 0. This implies that Q ⋄ must also be a block of P hom r . All of this shows that we may assume ℓ, ℓ ′ ∈ M\{0}. With the aid of (4.8) our assumption on ℓ, ℓ ′ may be written as m∈L K ℓ,m = m∈L K ℓ ′ ,m for all partition sets m∈L P m of P hom . (4.9) We will make use of the Krawtchouk coefficients for the case where m = e i = (0, . . . , 1, . . . , 0) (with 1 in the ith position). From Corollary 4.2 we have K ℓ,e i = K (n i ,q i ) 1 (ℓ i ) = (n i − ℓ i )q i − n i , which along with (4.9) results in the implication P e i is a block of P hom =⇒ ℓ i = ℓ ′ i . (4.10) Hence we aim at showing that the sets P e i are blocks of P hom . In order to do so, we assume without loss of generality that q 1 < . . . < q t . (4.11) Case 1: Suppose that for all i ∈ {1, . . . , t} we have q i − 1 = t j=1 (q j − 1) m j whenever t j=1 m j is odd. Then (4.7) shows that P e i is a block of P hom , and hence (4.10) implies ℓ = ℓ ′ . Then the same argument as in Case 1 along with (4.11) shows that the sets P ea , where a < i, are blocks of P hom . Hence by (4.10) ℓ a = ℓ ′ a for a = 1, . . . , i − 1. (4.12) Next, the set P ′ := m∈L P m , where L = {e i , m (1) , . . . , m (s) }, is a block of P hom . Since m (r) j = 0 for j ≥ i, the case m = ⋄ = ℓ in Corollary 4.2 along with (2.14) shows that K ℓ,m (r) = K (ℓ 1 ,...,ℓ i−1 ,0,...,0),m (r) for all r = 1, . . . , s and analogously for ℓ ′ . Hence K ℓ,m (r) = K ℓ ′ ,m (r) by (4.12) and thus m∈L K ℓ,m = s r=1 K ℓ,m (r) + K ℓ,e i = s r=1 K ℓ ′ ,m (r) + K ℓ,e i . Thus (4.9) yields K ℓ,e i = K ℓ ′ ,e i and hence ℓ i = ℓ ′ i . Now we may continue in the same fashion for the index set {i + 1, . . . , t}. If there is no index i ′ > i such that q i ′ − 1 = t j=1 (q j − 1) m j and t j=1 m j is odd, then we may argue as in Case 1. Otherwise, we choose the smallest i ′ > i and argue as in Case 2. Proceeding in this manner we finally arrive at ℓ = ℓ ′ , as desired.
We close the paper with the following summary.  n 1 ), . . . , (q t , n t )] is separating. In this case P hom r = P hom l . Moreover, P hom is self-dual if and only if R is semisimple and [(q 1 , n 1 ), . . . , (q t , n t )] is separating. (b) The homogeneous partition P hom on Z N is reflexive if and only if N is separating. The partition is self-dual if and only if N is square-free and separating.
We leave it to future research to determine the values and the corresponding partition of the homogeneous weight for general Frobenius rings.