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The kernel-block rank profile and a complete classification of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes

Dipak K. Bhunia

arXiv:2609.09969v1cs.ITmath.IT

TL;DR

Length, rank, and kernel dimension fail to distinguish two infinite families of pairs of distinct types. The kernel-block rank profile groups coordinates by their values across kernel words and records the dimensions of the span punctured on each block. ⌊(t^2+6)/12⌋ pairwise nonequivalent codes occur at length 2^t for every t≥3, and same-length family members are equivalent exactly when their types coincide.

  • Problem

    Length, rank, and kernel dimension fail to distinguish two infinite families of pairs of distinct types.

    Source: §1 · p. 3

  • Method

    The kernel-block rank profile groups coordinates by their values across kernel words and records the dimensions of the span punctured on each block.

    Source: §5 · p. 23

  • Results

    ⌊(t^2+6)/12⌋ pairwise nonequivalent codes occur at length 2^t for every t≥3, and same-length family members are equivalent exactly when their types coincide.

    Source: §1 · p. 4, §4.1 · p. 20

  • Takeaways & Limitations

    The profile separates the previously colliding types uniformly within the recursively constructed family, without computer equivalence tests.

    Source: §1 · p. 4

    The classification does not cover every Z2Z4Z8-additive Hadamard code of a given abstract type; classifying all such codes remains open.

    Source: §5 · p. 23

Abstract

from arXiv · show

The $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive codes are subgroups of $\mathbb{Z}_2^{\alpha_1}\times\mathbb{Z}_4^{\alpha_2}\times\mathbb{Z}_8^{\alpha_3}$, and a $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code is the Gray map image of such a code. A recursive construction of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive Hadamard codes $\mathcal H^{t_1,t_2,t_3}$, with all $\alpha_i\neq0$, $t_1\geq1$, $t_2\geq0$, and $t_3\geq1$, is known, as are the linearity, kernel dimension, and rank of the corresponding codes $H^{t_1,t_2,t_3}$ of length $2^t$, where $t+1=3t_1+2t_2+t_3$. Yet these invariants do not completely classify the family. Two infinite families of pairs of distinct types share the length, rank and kernel dimension, and for classified lengths $2^t$, $3\leq t\leq11$, such pairs were separated only by computer equivalence tests. In this paper, we introduce an equivalence invariant that resolves these cases. The kernel partitions the binary coordinates into blocks, two coordinates lying in the same block when every kernel word takes the same value in both; the \emph{kernel-block rank profile} is the multiset of the dimensions of the linear span punctured on these blocks. Unlike rank and kernel dimension, which are global, this invariant records how much of the span survives on each block. We compute it for the whole family: there are $2^{t_1+t_2+t_3-1}$ blocks, all of size $2^{2t_1+t_2}$, and the profile takes at most two values $t_2+\binom{t_1+2}{2}$ and $t_2+2+\binom{t_1+1}{2}$, whose difference is $t_1-1$; it is constant precisely when $t_1=1$. Hence, the profile recovers $t_1$, and the length and kernel dimension recover $t_2$ and $t_3$. Two codes of the family with the same length are therefore equivalent if and only if their types coincide, and the number of pairwise nonequivalent such codes of length $2^t$ is $\lfloor(t^2+6)/12\rfloor$ for every $t\geq3$.

1 Introduction

The recursively constructed Z2Z4Z8-linear Hadamard family is not classified by rank and kernel dimension alone. The paper introduces the kernel-block rank profile and uses it to distinguish types and classify the family internally.

1 Introduction

  • Two infinite families of distinct types share length, rank, and kernel dimension, while earlier computer tests separated such pairs only for lengths 2^t with 3≤t≤11.The rank and kernel dimension are global invariants and do not record where vectors occur among coordinates.Source: §1 · p. 3
  • The kernel-block rank profile partitions coordinates by their values across kernel words, then records the span dimension surviving on each block.This multiset is an equivalence invariant.Source: §1 · pp. 3–4
  • The family’s kernel partition has 2^(t1+t2+t3−1) equal-sized blocks, and the profile has at most two values whose difference is t1−1.The profile is constant precisely when t1=1.Source: §1 · p. 4
  • The profile recovers t1, while length and kernel dimension recover t2 and t3, so same-length family members are equivalent exactly when their types coincide.There are ⌊(t^2+6)/12⌋ pairwise nonequivalent codes of length 2^t for every t≥3.Source: §1 · p. 4
  • The classification is internal to the recursively constructed family; equivalence with other Hadamard-code families is not addressed.Comparisons with Z4-linear, Z2Z4-linear, or Z2s-linear families require their profiles.Source: §1 · p. 4

2 Preliminaries

This section establishes the tools and framework for analyzing the recursive Hadamard family, including puncturing, Boolean coordinate functions, and digit formulas for rank. It also records how rank varies with t3 and identifies types that rank and kernel dimension alone cannot distinguish.

2.1 Puncturing and equivalence

  • Puncturing to positions B measures the rank visible there, while code equivalence by translation and coordinate permutation carries the span and kernel by that permutation.In particular, equivalent codes have the same rank and kernel dimension.Source: §2.1 · p. 4

2.2 Generator rows, additive coordinates and binary coordinates

  • For binary coordinates, punctured rank equals the dimension of the span of their Boolean coordinate functions, and Gray-output functions at an additive coordinate span its binary digits.For a family of functions polynomial in free binary parameters, the span of the family equals the span of its coefficient functions.Source: §2.2 · p. 6

2.3 The recursive family Ht1,t2,t3

  • The recursive family yields binary Hadamard codes of length 2^t, where t+1=3t1+2t2+t3; nonlinear members have kernel dimension t1+t2+t3.The construction adds rows of orders 8, 4, and 2, with the first row distinguished and of order 2.Source: §2.3 · pp. 7–8

2.4 Message digits, coordinate digits and the rank

  • Computations reduce to t3=1 and use binary carry identities to express additive-coordinate digits; for a Z4 coordinate, its two Gray bits span the functions P and h.As the coordinate ranges over the Z4 part, its parameters run over the stated normalized admissible quadruples.Source: §2.4 · pp. 8–9
  • Binary digits of additive coordinates are obtained by summing column entries against message coefficients, with symmetric-polynomial terms accounting for carries.The higher coordinate digits are expressed in terms of the free digits of the additive coordinate.Source: §2.4 · p. 10
  • The Z2 coordinates contribute the span of the constant function and the message digits a_i and d_j.The coordinate function is ϵ + α_S + δ_R as S and R vary.Source: §2.4 · p. 10
  • Increasing t3 by one increases rank(H^t1,t2,t3) by one.Source: §2.4 · p. 10
  • Distinct-type nonlinear family members can share length, rank, and kernel dimension; the proposition states that each listed pair has these equalities.The supplied passage does not include the complete list of pair types.Source: §2.4 · p. 10

3 The kernel-block rank profile

This section develops the kernel-block rank profile by identifying kernel-defined coordinate blocks and computing the dimensions of the linear span restricted to them, first for nonlinear codes H^{t1,t2,1} and then for the full family. It assembles these local ranks into a profile that records the blocks and their rank values.

3.1 Definition and invariance

  • The kernel-block rank profile is the multiset of binary-span dimensions punctured on blocks whose coordinates agree in every kernel word, and it is preserved by code equivalence.Under an equivalence given by a coordinate permutation and translation, the permutation maps kernel blocks bijectively and preserves each block’s local rank.Source: §3 · p. 11, §3.1 · p. 11

3.2 The kernel labels and the blocks

  • For nonlinear H^{t1,t2,1}, kernel labels determine 2^(t1+t2) nonempty blocks, each containing 2^(2t1+t2) binary coordinates.Blocks with nonzero first label component come from the Z8 part; those with zero first and nonzero second component come from the Z4 part, while the zero label combines Z2 coordinates with some Z4 coordinates.Source: §3.2 · p. 13

3.3 The local ranks

  • On the zero-label block, the local rank is t2+1+2t1+binom(t1,2); blocks with zero first and nonzero second label have the same local rank.The zero-label punctured span is generated by distinct square-free monomials; the other blocks attain the same rank through corrected functions depending on their label.Source: §3.3 · pp. 15–16
  • For H2,0,1, the four blocks comprise one block of local rank 6 and three of rank 5, giving profile {{6[1], 5[3]}}.The local ranks are obtained by spanning the punctured coordinates with listed functions and proving their linear independence.Source: §3.3 · pp. 16–17

3.4 The profile of Ht1,t2,t3

  • Under Plotkin extension, each kernel block yields two same-sized blocks with unchanged local ranks, doubling every profile multiplicity.The extension also preserves the all-one kernel word, allowing the construction to be iterated for increasing t3.Source: §3.4 · p. 18
  • The kernel partition has 2^(t1+t2+t3−1) blocks, all of size 2^(2t1+t2); moreover, ρ0−ρ1=t1−1, so the profile is constant exactly when t1=1.Source: §3.4 · p. 19
  • For H2,0,2, the profile becomes {{6[2], 5[6]}}, distinguishing it from constant-profile family members despite the limited information in global rank and kernel dimension.The example contrasts this profile with H1,1,1, whose four blocks all have local rank 4.Source: §3.4 · p. 19

4 The complete classification inside the family

The kernel-block rank profile recovers the code parameters and yields a complete classification within this family: equal-length codes are equivalent exactly when their types match. The resulting class count is established for every t, while the profile also separates the collision families that rank and kernel dimension miss.

4 The complete classification inside the family

  • Rank and kernel dimension leave collisions unresolved: earlier classifications used computer equivalence tests for cases through lengths 2^11.For length 2^9, H1,2,3 and H2,0,4 share (r, k) = (15, 6).Source: §4 · p. 20

4.1 Recovering the parameters

  • The profile recovers t1, and length plus kernel dimension then recover t2 and t3.For nonlinear members with t1 ≥ 2, the profile alone determines the type; the unique linear member at each length is handled separately.Source: §4 · p. 20, §4.1 · pp. 20–21
  • At every length, two family members are equivalent if and only if their types coincide.Thus distinct admissible types give nonequivalent codes.Source: §4.1 · p. 20

4.2 The number of classes and the two collisions

  • The number of equivalence classes equals the number of admissible types, a count now proved for every t ≥ 3.The result agrees with earlier Magma counts for 3 ≤ t ≤ 11.Source: §4.2 · p. 21
  • The profile distinguishes both collision families throughout their stated ranges, so the paired codes are nonequivalent.For the first pair, one profile is constant and the other is not; for the second, their profile values differ.Source: §4.2 · p. 21
  • Table 1 lists rank, kernel dimension, and profile for lengths 2^t with 5 ≤ t ≤ 12; its profiles make nonequivalence visible without further tests.At t = 9, the profiles of H1,2,3 and H2,0,4 are respectively constant and nonconstant.Source: §4.2 · p. 22, Table 1 · p. 22

5 Conclusions and further research

The kernel-block rank profile classifies the recursively constructed family internally by recovering its type from the profile, length, and kernel dimension. The paper also identifies boundaries on that classification and directions for extending the method.

5 Conclusions and further research

  • Same-length codes in the recursively constructed family are equivalent exactly when their types coincide, yielding floor((t^2+6)/12) equivalence classes at length 2^t for t≥3.The profile recovers t1, while length and kernel dimension recover t2 and t3.Source: §5 · p. 23
  • The kernel-block rank profile takes at most two values differing by t1−1, recovering t1 for this family.The kernel partitions coordinates into equal-sized blocks, and the profile records the span dimensions punctured on those blocks.Source: §5 · p. 23
  • The classification covers the recursively constructed family, not every code of its abstract type; an alternative example has rank 13 rather than H2,0,1’s rank 12.Classifying all Z2Z4Z8-additive Hadamard codes of a given abstract type remains open.Source: §5 · p. 23
  • Comparison with previously known Hadamard-code families remains incomplete because their kernel-block rank profiles have not been computed; t1=1 needs further argument.For t1≥2, nonconstant profiles would separate these codes once the comparison profiles are known.Source: §5 · p. 23
  • A proposed hierarchy records span dimensions on unions of ℓ kernel blocks, but whether its ℓ=2 refinement is strictly finer remains open.The paper also leaves open how much of an arbitrary additive code’s structure the full hierarchy recovers.Source: §5 · p. 23
  • Extensions to other mixed alphabets require recomputing labels and local ranks; Z4Z8-additive Hadamard-code existence remains open, while α1≠0, α2=0, α3≠0 cannot occur.The invariant’s definition and a lemma apply unchanged in the proposed broader settings.Source: §5 · p. 23
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