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The kernel-block rank profiles of the $\mathbb{Z}_2\mathbb{Z}_4$-linear and the $\mathbb{Z}_{2^s}$-linear Hadamard codes, and a complete classification of the $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes

Dipak K. Bhunia

arXiv:2610.00142v2cs.ITmath.IT

TL;DR

Most Hadamard codes are nonlinear, and their classification remains open; rank and kernel dimension alone do not always distinguish codes. This paper computes kernel-block rank profiles for comparison families and proves a descent result for the Z2^s-linear family. The resulting profile separates the nonlinear Z2Z4Z8-linear codes from same-length comparison codes, with a two-block refinement resolving the remaining shared-profile family.

  • Problem

    Most Hadamard codes are nonlinear and their classification remains open, while rank and kernel dimension do not always distinguish codes.

    Source: §1 · pp. 2–3

  • Method

    The paper computes kernel-block rank profiles for comparison families, proves a descent theorem for Z2^s-linear codes, and uses a two-block refinement for the residual family.

    Source: §6.3 · p. 25, §7 · p. 31

  • Results

    No nonlinear Z2Z4Z8-linear Hadamard code is equivalent to a same-length Z4-, Z2Z4-, or Z2^s-linear Hadamard code; the residual family has µ equal to 6 on one side and 7 on the other.

    Source: §1 · p. 4

  • Takeaways & Limitations

    The profile and its two-block refinement complete the comparison-classification result for the recursively constructed Z2Z4Z8-linear Hadamard family.

    Source: §1 · p. 4, §7 · pp. 31–32

    The results concern the recursively constructed Z2Z4Z8-additive family; classifying all codes of a given abstract type remains open.

    Source: §7 · p. 32

Abstract

from arXiv · show

The kernel of a binary code containing the zero word partitions the binary coordinates into blocks, two coordinates lying in the same block when every kernel word takes the same value in both, and the \emph{kernel-block rank profile} is the multiset of the dimensions of its linear span punctured on those blocks. For the family $H^{t_1,t_2,t_3}$ of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes, this invariant is known explicitly and gives a complete classification of the family. In this paper, we compute it for the $\mathbb{Z}_2\mathbb{Z}_4$-linear and $\mathbb{Z}_{2^s}$-linear Hadamard families with which those codes are compared, and we prove that it is constant for every nonlinear $\mathbb{Z}_2\mathbb{Z}_4$-linear Hadamard code and for every nonlinear $\mathbb{Z}_{2^s}$-linear Hadamard code $\bar H^{a_1,\dots,a_s}$ with $s\geq2$. The second statement is obtained without any rank formula, by exhibiting coordinate permutations that preserve the code and act transitively on its kernel blocks, which makes the argument uniform in $s$. We also prove a descent theorem: if $\bar H^{a_1,\dots,a_s}$ is nonlinear and $a_1\geq2$, then the code punctured on one kernel block is the $\mathbb{Z}_{2^{s-1}}$-linear Hadamard code $\bar H^{a_1,\dots,a_{s-1}}$. Consequently, the constant local rank equals $rank(\bar H^{a_1,\dots,a_{s-1}})$ and is at least $t-\kappa+2$, where $2^t$ is the length and $\kappa$ the kernel dimension. These results separate every nonlinear $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code from every $\mathbb{Z}_4$-linear, $\mathbb{Z}_2\mathbb{Z}_4$-linear and $\mathbb{Z}_{2^s}$-linear Hadamard code of the same length, except for the single infinite family $H^{1,1,t-4}$ and $\bar H^{2,0,t-5}$ with $t\geq5$. The members of this infinite family agree in the rank, kernel dimension and the kernel-block rank profile, but a two-block refinement separates them.

1 Introduction

The paper completes the classification comparisons for recursively constructed Z2Z4Z8-linear Hadamard codes by computing the kernel-block rank profiles of the families used for comparison. It also proves a puncturing descent result and resolves the remaining exceptional comparison with a two-block refinement.

1 Introduction

  • Rank and kernel dimension are equivalence invariants, but matching values do not generally establish equivalence, complicating classification.Hadamard codes are often nonlinear, and their classification remains open.Source: §1 · pp. 2–3
  • Earlier work classified Z4-linear and Z2Z4-linear Hadamard codes using rank or kernel dimension, while rank formulas are unavailable for Z2s-linear codes when s ≥ 4.The paper positions profile computation as a way to compare these families with Z2Z4Z8-linear Hadamard codes.Source: §1 · p. 3
  • The kernel-block rank profile was previously computed for the recursively constructed Z2Z4Z8-linear family and classifies that family internally.For codes of the same length, the cited result says equivalence holds exactly when their types coincide.Source: §1 · p. 3
  • Nonlinear Z2Z4-linear Hadamard codes have constant kernel-block rank profile, and every nonlinear Z2s-linear Hadamard code has a constant profile for s ≥ 2.For the latter family, coordinate permutations act transitively on kernel blocks, so the proof uses no rank formula and works uniformly in s.Source: §1 · p. 4
  • For nonlinear Z2s-linear Hadamard codes with a1 ≥ 2, puncturing on one kernel block gives the Z2^(s−1)-linear Hadamard code one alphabet lower.The common local rank is the rank of that punctured code and is at least t − κ + 2.Source: §1 · p. 4
  • The comparison proves that no nonlinear Z2Z4Z8-linear Hadamard code is equivalent to a same-length Z4-linear, Z2Z4-linear, or Z2s-linear Hadamard code.A two-block refinement separates the exceptional family whose members share rank, kernel dimension, and the full profile.Source: §1 · p. 4

2 Preliminaries

The preliminaries define the kernel-block rank profile and introduce the Boolean-function tools and code-family constructions used in later comparisons. They also review the relevant Hadamard-code families, their kernel structure and representation overlaps, and the known profile, linearity criterion, and limits of rank formulas.

2.1 Puncturing and the kernel-block rank profile

  • The kernel-block rank profile is the multiset of local ranks, where coordinates share a block if every kernel word agrees on them, and each local rank is the dimension of the punctured span [23].The profile records multiplicities, not just distinct rank values; it is constant when all entries are equal.Source: §2.1 · p. 5

2.2 Boolean functions attached to binary coordinates

  • Local rank on a coordinate set equals the dimension of the span of its Boolean coordinate functions, while binary digits of sums are expressed by elementary symmetric Boolean polynomials [33].Distinct square-free monomials in independent binary variables are linearly independent, enabling the dimension counts used later.Source: §2.2 · p. 6

2.3 The Z2Z4Z8-linear Hadamard codes and their profile

  • A Z2Z4Z8-linear code is linear exactly when (t1,t2)=(1,0); for nonlinear codes, the kernel is the Gray image of the subcode of elements of order at most two [33].Every Z4-linear or Z2Z4-linear code sharing rank and kernel dimension with a nonlinear H^{t1,t2,t3} is equivalent to H^{5,t−9} when (t1,t2)=(2,1), for t≥10 [33].Source: §2.3 · pp. 7–8
  • For nonlinear H^{t1,t2,t3}, the kernel partition has 2^(t1+t2+t3−1) blocks of size 2^(2t1+t2), and its profile is constant exactly when t1=1 [23].The two local ranks satisfy ρ0−ρ1=t1−1.Source: §2.3 · p. 8

2.4 The Z2Z4-linear Hadamard codes

  • The Z2Z4-linear Hadamard family is obtained by deleting the Z8 part of the construction, and every Z4-linear Hadamard code is equivalent to a member with U≥2 and V≥1.The family is linear exactly when U=1; its nonlinear kernels are Gray images of the subcodes of elements of order at most two.Source: §2.4 · pp. 8–9

2.5 The Z2^s-linear Hadamard codes

  • The Z2^s-linear construction begins with parameters a1,…,as, where a1≥1, and a matrix whose rows are indexed by their sum m=a1+…+as [the supplied passage ends mid-definition].The passage specifies T1=Z2^s and describes each Ti as an ideal whose elements have order at most 2^(s−i+1).Source: §2.5 · p. 9

2.5 The Z2s-linear Hadamard codes

  • Every Z2^s-additive Hadamard code is permutation equivalent to a code in the stated parameterized family.The family members are binary Hadamard codes; their binary coordinates can be indexed by pairs consisting of an additive coordinate and a vector in Z2^(s−1).Source: §2.5 · p. 10
  • For s ≥ 3, linearity holds exactly for parameter tuples (1, 0, …, 0, a_s) or (1, 0, …, 0, 1, a_s); for s = 2, it holds exactly when a_1 ≤ 2.The nonlinear codes’ kernel basis includes Gray images of order-two generators and constant words.Source: §2.5 · p. 10
  • No rank formula is known for s ≥ 4, while the rank for s = 3 was determined in.This lack of a formula is why the comparison in [33] stops at Z8.Source: §2.5 · p. 10
  • If a_s ≥ 1, each 1 ≤ ℓ ≤ a_s permits a permutation-equivalent representation over Z2^(s+ℓ), raising the alphabet by ℓ levels.Read in reverse, the theorem permits lowering the alphabet when the first parameter equals 1.Source: §2.5 · p. 10
  • For a nonlinear Z2^s-linear Hadamard code, 1 ≤ σ ≤ s − 1.When a_1 ≥ 2, σ = 1; when a_1 = 1, nonlinearity requires a positive parameter at some level from 2 through s − 1, giving the upper bound.Source: §2.5 · p. 11

3 The profile of the Z2Z4-linear Hadamard codes

The section computes the kernel-block rank profile of nonlinear Z2Z4-linear Hadamard codes: every block has the same size and local rank. The computation uses kernel labels and the spans of coordinate functions, then extends the result by duplication.

3 The profile of the Z2Z4-linear Hadamard codes

  • The profile is computed by labeling coordinates from kernel words, counting coordinates per label, and finding the span of their coordinate functions.The computation is shorter than for the Z2Z4Z8 family because there are no rows of order 8.Source: §3 · pp. 11–12
  • Every nonlinear HU,V has 2^(U+V−1) kernel blocks, each of size 2^U and local rank U+1.Thus its kernel-block rank profile is constant.Source: §3 · pp. 11–13
  • For general V, each duplication step doubles the number of blocks while preserving block size and local rank.The V=1 case has 2^U blocks, each of size 2^U and local rank U+1.Source: §3 · pp. 12–13

4 The profile of the Z2s-linear Hadamard codes

For nonlinear Z2^s-linear Hadamard codes, the kernel-block rank profile is constant for every s≥2, even though no rank formula is available for s≥4. The proof constructs code-preserving permutations that act transitively on the blocks.

4.1 Two families of permutations preserving the code

  • Translations of the additive coordinate and shifts of the Gray index give two families of coordinate permutations preserving the code.Only σ Gray-index shifts are shown to preserve the code; the additive-coordinate translations are indexed by δ∈G.Source: §4 · p. 13, §4.1 · pp. 13–14

4.2 The kernel blocks, and transitivity

  • The kernel blocks are labeled by parity data from additive coordinates and a Gray-index label, yielding 2^(ker(H̄)−1) blocks.The parity labels range independently, and the Gray-index label is determined by inner products with a σ-dimensional space.Source: §4.2 · pp. 15–16
  • The two permutation families act transitively on kernel blocks, making any two punctured block codes permutation equivalent and giving them the same local rank.The theorem thereby establishes the constant profile for every nonlinear code in the family.Source: §4.2 · pp. 16–17

4 The profile of the Z2s-linear Hadamard codes

  • For every nonlinear Z2^s-linear Hadamard code with s≥2, all kernel blocks have the same size and local rank, so the profile is constant.The proof obtains constancy without computing any local rank.Source: §4 · p. 13, §4.2 · p. 17

5 The punctured code on a kernel block

When a nonlinear Z2^s-linear Hadamard code has a1≥2, puncturing on a kernel block produces the Z2^(s−1)-linear Hadamard code with the same parameters up to the last alphabet level. This identifies the common local rank and yields a sharp lower bound.

5 The punctured code on a kernel block

  • If H̄a1,…,as is nonlinear and a1≥2, puncturing on a kernel block gives the Z2^(s−1)-linear Hadamard code H̄a1,…,as−1.The theorem identifies the punctured code via a coordinate bijection from the block to the lower-alphabet code’s coordinates.Source: §5 · pp. 17–18
  • The common local rank is rank(H̄a1,…,as−1), independent of as, and is at least t−κ+2.Equality holds exactly when s=2, or s=3 and a1=2.Source: §5 · p. 20
  • Each kernel block has 2^(t−κ+1) binary coordinates.There are 2^(κ−1) blocks, all of equal size.Source: §5 · p. 20
  • For nonlinear Z2Z4Z8-linear Hadamard codes, the profile is constant and equals t−κ+2 exactly when t1=1.The comparison also gives ρ1=t−κ+2 exactly when t1≤2.Source: §5 · p. 20
  • The a1≥2 condition loses no nonlinear cases, since repeated parameter reduction gives an equivalent code whose first parameter is at least two.The reduction stops at s′=2 only for codes equivalent to nonlinear Z4-linear Hadamard codes.Source: §5 · p. 21

6 The complete classification

The section completes the equivalence classification of [?]Z₂Z₄Z₈-linear Hadamard codes by separating cases with t₁ ≥ 2 through nonconstant profiles and cases with t₁ = 1 through profile and rank comparisons. Kernel-block rank profiles distinguish the remaining undecided families, while a two-block invariant resolves the exceptional pair; μ is needed for only one family.

6.1 The codes with t1 ≥2

  • Every Z2Z4Z8-linear Hadamard code with t1 ≥2 is inequivalent to same-length Z4-linear, Z2Z4-linear, and Z2^s-linear Hadamard codes.Its profile is nonconstant, while the comparison families have constant profiles.Source: §6.1 · p. 22

6.2 The codes with t1 = 1

  • For nonlinear codes with t1 = 1, profile comparisons exclude all Z2^s-linear candidates except the extremal cases, which require rank comparisons.The profile value is t − κ + 2; a comparison profile is larger except when s′ = 2 or s′ = 3 and a1 = 2.Source: §6.2 · pp. 22–23
  • Nonlinear H1,t2,t3 codes are nonequivalent to same-length Z4-linear and Z2Z4-linear Hadamard codes, and to nonlinear Z2^s-linear Hadamard codes (s≥2), except when t2=1 and the latter is permutation equivalent to ¯H2,0,t−5.Source: §6.2 · p. 23
  • The remaining pair H1,1,t−4 and H̄2,0,t−5 has the same rank, kernel dimension, and kernel-block profile, so those invariants do not distinguish it.For t ≥ 5, both have kernel dimension t − 2, rank t + 3, and local rank 4 on every block.Source: §6.2 · p. 24

6.3 A two-block refinement

  • The equivalence-invariant two-block refinement distinguishes the remaining pair: at length 32, μ(H1,1,1) = 6 and μ(H̄2,0,0) = 7.It takes the maximum punctured-span dimension over unions of two distinct kernel blocks; its behavior under Plotkin extension reduces the infinite family to its shortest member.Source: §6.3 · pp. 24–26
  • For every t ≥5, µ separates H1,1,t−4 from ¯H2,0,t−5, resolving the exceptional family that agrees in rank, kernel dimension, and kernel-block rank profile.The six two-block ranks of ¯H2,0,0 are 5, 5, 7, 7, 7, 7, giving µ = 7.Source: §6.3 · pp. 27–28

6.4 The classification theorem

  • For every t ≥3, the ⌊(t^2 + 6)/12⌋ codes Ht1,t2,t3 are pairwise nonequivalent, and all but the unique linear code differ from the comparison families.The comparison families are Z4-linear, Z2Z4-linear, and Z2^s-linear Hadamard codes for s ≥2.Source: §6.4 · p. 28
  • When t1 ≥2, the Z2Z4Z8-linear profile is nonconstant, whereas the two comparison families have constant profiles; their coarser block data agree.The shared coarser data are 2^κ−1 blocks of size 2^(t−κ+1).Source: §6.4 · p. 28, Table 1 · p. 28
  • Among the six infinite families left undecided by rank and kernel dimension, only the second requires µ; the other five are separated by their kernel-block rank profiles.For example, H2,1,t−7 has profile {{7[2^(t−7)], 6[3·2^(t−7)]}}, unlike the constant profile of H5,t−9.Source: §6.4 · pp. 30–31

7 Conclusions and further research

The paper computes kernel-block rank profiles for the comparison families and uses them to complete classification of the recursively constructed Z2Z4Z8-linear Hadamard family. A descent theorem links local ranks to ranks of lower-alphabet codes, while open rank and classification questions delimit the results.

Comparison profiles

  • Nonlinear Z2Z4-linear Hadamard codes HU,V have 2^(U+V)-1 kernel blocks of size 2^U and constant local rank U+1.Source: §7 · p. 31
  • For every nonlinear Z2s-linear Hadamard code with s≥2, coordinate permutations act transitively on kernel blocks, proving a constant profile without a rank formula.The argument is uniform in s; the admissible shifts are governed by parameter σ.Source: §7 · p. 31

Classification

  • The comparison completes classification except H^{1,1,t-4} and Hbar^{2,0,t-5} for t≥5, which share rank, kernel dimension, and profile but are separated by the two-block invariant µ.The rank settles the other extremal cases.Source: §7 · p. 31

Descent and rank bound

  • For nonlinear Hbar^{a1,...,as} with a1≥2, puncturing one kernel block gives Hbar^{a1,...,as−1}, so the common local rank equals that lower-alphabet code’s rank.The local rank is at least t−κ+2, with equality exactly for s=2 or for s=3 and a1=2.Source: §7 · p. 31

Further research

  • For s−1≥4, the rank of Hbar^{a1,...,as−1} remains open; descent makes that rank a local quantity computable on one kernel block.That block is smaller than the whole code.Source: §7 · p. 31
  • The rank-on-unions hierarchy’s second level separates the exceptional family, but how much structure the full hierarchy recovers remains an open question.The paper asks whether some ℓ suffices to classify a given family.Source: §7 · pp. 31–32
  • The results concern the recursively constructed Z2Z4Z8-additive Hadamard family; classifying all codes of a given abstract type remains open.Generalizations to other alphabets and the existence of Z4Z8-additive Hadamard codes are also open.Source: §7 · p. 32
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