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Rank and classification of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes

Dipak K. Bhunia

arXiv:2609.06244v1cs.ITmath.IT

TL;DR

The rank of the recursively constructed Z2Z4Z8-linear Hadamard codes was not known in closed form, leaving their classification by rank and kernel dimension unresolved for arbitrary types. This paper derives a rank formula by studying binary coordinate functions as Boolean functions of message digits. The formula and known kernel dimensions specify how far these invariants classify the family and identify its remaining coincidences.

  • Problem

    Although the codes’ linearity and kernel were known, no general rank formula existed to determine how far rank and kernel dimension distinguish the family for arbitrary types.

    Source: §1 · pp. 2–3

  • Method

    The paper computes rank as the dimension of the span of binary coordinate functions, treated as Boolean functions of the message digits.

    Source: §3 · p. 7

  • Results

    For length 2^t, (r, k) distinguishes floor((t^2+6)/12) classes for 3 ≤ t ≤ 6, one fewer for 7 ≤ t ≤ 9, and two fewer for t ≥ 10.

    Source: §1 · p. 4

  • Takeaways & Limitations

    Rank and kernel dimension classify the family completely only for 3 ≤ t ≤ 6; the paper also identifies one infinite coincidence with the Z4- and Z2Z4-linear families and five with the Z8-linear family.

    Source: §1 · p. 4, §5 · p. 31

    The conclusions concern the recursively constructed family and constructions proved equivalent to it; comparisons with Z2s-linear Hadamard codes cover only s ≤ 3.

    Source: §5 · p. 31

Abstract

from arXiv · show

The $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive codes are subgroups of $\mathbb{Z}_2^{\alpha_1}\times\mathbb{Z}_4^{\alpha_2}\times\mathbb{Z}_8^{\alpha_3}$. A $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code is a Hadamard code which 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 and kernel of the corresponding codes $H^{t_1,t_2,t_3}$ of length $2^t$. Their rank, however, was not known in closed form. In this paper, we determine $rank(H^{t_1,t_2,t_3})$ for every admissible triple and obtain an explicit closed formula. The proof computes the dimension of the space spanned by the binary coordinate functions of the code, viewed as Boolean functions of the binary digits of the message. We then use the rank formula, together with the known kernel dimension, to classify the family as far as these two invariants allow. The only pairs of distinct types of the same length sharing both invariants are $\bigl(H^{1,2,t-6},H^{2,0,t-5}\bigr)$ for $t\geq7$ and $\bigl(H^{2,2,t-9},H^{3,0,t-8}\bigr)$ for $t\geq10$. Consequently, the number of distinct pairs $(r,k)$ among the codes of length $2^t$ is $\lfloor(t^2+6)/12\rfloor$ for $3\leq t\leq6$, $\lfloor(t^2+6)/12\rfloor-1$ for $7\leq t\leq9$, and $\lfloor(t^2+6)/12\rfloor-2$ for $t\geq10$. Thus, rank and kernel dimension classify the family completely if and only if $3\leq t\leq6$. We also determine all coincidences with the $\mathbb{Z}_4$-, $\mathbb{Z}_2\mathbb{Z}_4$-, and $\mathbb{Z}_8$-linear Hadamard codes of the same length: there is exactly one infinite family with the first two, namely $H^{2,1,t-7}$ and $H^{5,t-9}$ for $t\geq10$, and exactly five families with the $\mathbb{Z}_8$-linear codes.

1 Introduction

The rank of the recursively constructed Z2Z4Z8-linear Hadamard codes was unknown in closed form, limiting what rank and kernel dimension could establish about their classification. This paper derives the rank and uses both invariants to describe the family’s remaining coincidences, including comparisons with classical families.

1 Introduction

  • The rank of Ht1,t2,t3 was not known in general, leaving its classification by rank and kernel dimension without a theoretical account for arbitrary types.Earlier rank values were computed type by type with Magma for lengths 2^t with 3 ≤ t ≤ 11.Source: §1 · p. 3
  • The rank formula comes from the dimension of the span of binary coordinate functions viewed as Boolean functions of message digits.The Z2 and Z4 coordinates span V0; the lowest two digits of Z8 coordinates already lie in V0, so only the top digit can add rank.Source: §1 · pp. 3–4
  • The formula agrees with every rank value previously computed with Magma.The comparison is with the values in [33, Tables 2–4].Source: §1 · p. 4
  • For length 2^t, the pair (r, k) distinguishes floor((t^2+6)/12) classes for 3 ≤ t ≤ 6, one fewer for 7 ≤ t ≤ 9, and two fewer for t ≥ 10.Thus rank and kernel dimension classify the family completely exactly for 3 ≤ t ≤ 6.Source: §1 · p. 4
  • Against the Z4- and Z2Z4-linear families, exactly one infinite coincidence remains; against the Z8-linear family, five coincidence families remain.The single family with the first two is H2,1,t−7 and H5,t−9 for t ≥ 10.Source: §1 · p. 4

2 Preliminaries and the recursive family

This section presents the recursively constructed family, a direct description of its generator-matrix columns, and previously established results on its Hadamard property, linearity, and kernel.

2.1 The recursive construction of At1,t2,t3

  • Starting from A1,0,1, the recursion adds t1−1 rows of order 8, t2 rows of order 4, and t3−1 rows of order 2.The first row has order 2 and is distinguished in the construction.Source: §2.1 · pp. 4–5

2.2 A direct description of the columns of At1,t2,t3

  • For proofs, the recursive matrices are replaced by a direct column description: the Z2 columns range over all vectors, Z4 columns are normalized primitive vectors, and Z8 columns pair normalized primitive q with arbitrary 2r entries.For t3 > 1, the matrix is obtained by repeating the specified duplication on the t3=1 construction.Source: §2.2 · p. 6

2.3 Linearity and kernel of Ht1,t2,t3

  • The construction yields a binary Hadamard code of length 2^t, where t+1=3t1+2t2+t3.Its linear members are exactly H1,0,t3; for nonlinear members, the kernel dimension is t1+t2+t3.Source: §2.3 · p. 7

3 The rank of Ht1,t2,t3

This section derives a closed rank formula for every admissible parameter triple by analyzing the span of the binary coordinate functions, including its base and carry components. It checks the formula against the known linear Hadamard dimension and the kernel dimensions of nonlinear members.

3 The rank of Ht1,t2,t3

  • A closed formula determines rank(H^{t1,t2,t3}) for all t1 ≥ 1, t2 ≥ 0, and t3 ≥ 1.The formula addresses the mixed-alphabet family, for which no closed formula was previously available.Source: §3 · p. 7

3.1 Generator rows, additive coordinates and binary coordinates

  • The rank is computed as the dimension of the span of binary coordinate functions, reducing the problem to Boolean functions of the message digits.The computation finds the span for t3 = 1 and extends the result to arbitrary t3 using duplication (5).Source: §3 · p. 7

3.3 The three binary digits of a coordinate in the Z8 part

  • For Z8 coordinates, the bottom two binary digits lie in V0, so only the top digit can increase rank.The digit expansions involve carries; for sums of bits, the binary digits are given by the elementary symmetric functions sym1, sym2, and sym4.Source: §3 · p. 7, §3.3 · pp. 11–12

3.2 The space V0 coming from the Z2 and the Z4 parts

  • The Z2- and Z4-part coordinate functions span V0, with basis the lowest message digits and pairwise products of distinct elements of L.Here L consists of the bottom digits a_i and d_j; the products represent carries from adding bottom digits.Source: §3.2 · p. 10

3.3 The three binary digits of a coordinate in the Z8 part

  • For t3 = 1, the binary coordinate functions span V0 + W, so rank(H_t1,t2,1) = dim(V0 + W).Source: §3.3 · pp. 12–13

3.4 The reduced top digit

  • Modulo V0, each top digit is determined by nonempty S and a linear form ℓ; every such pair is realized, and for t1 = 1 the contribution has dimension 1 + t2.Source: §3.4 · pp. 13–14

3.5 The additional carry space

  • The additional carry space splits into disjoint parts W1 and W2 with monomial bases; for t1 = 2 and t2 = 0, its dimension is 6.Source: §3.5 · pp. 15–16

3.6 The rank formula

  • For every admissible triple, Theorem 3.15 gives a closed-form rank; for t3 > 1, the rank is rank(H_t1,t2,1) + (t3 − 1).Examples include rank(H_1,1,1) = 8 and rank(H_2,0,1) = 12.Source: §3.6 · p. 17
  • For (t1, t2) = (1, 0), rank(H1,0,t3) = t3 + 3 = t + 1, matching the dimension of the binary linear Hadamard code of length 2^t.Here t = t3 + 2.Source: §3.6 · p. 18
  • In the linear case, the kernel dimension is t + 1, not t1 + t2 + t3, explaining why the theorem’s kernel statement is restricted to nonlinear members.Source: §3.6 · p. 18
  • For every nonlinear member, formula (43) gives rank(Ht1,t2,t3) > ker(Ht1,t2,t3) = t1 + t2 + t3, as required for a nonlinear code.Source: §3.6 · p. 18

4 Classification by the rank and the dimension of the kernel

This section uses rank and kernel dimension to classify the family, identify within-family collisions, and compare its codes with other Hadamard families. It also describes coincidences with -linear, -linear, and -linear Hadamard codes, while noting limitations of these comparisons.

4 Classification by the rank and the dimension of the kernel

  • For nonlinear Z2Z4Z8-linear codes, fixing length and kernel dimension determines the parameter triple except for t1.The remaining question is whether rank distinguishes choices of t1.Source: §4 · p. 18

4.1 The comparison families

  • The comparisons include Z4-linear codes through their equivalence to Z2Z4-linear codes, and Z8-linear codes only for the rank-known s = 3 case.The paper notes that rank comparisons are unavailable for Z2^s-linear alphabets with s ≥ 4 because no rank formula is available.Source: §4.1 · p. 19

4.2 The reduced rank

  • The reduced ranks of the three families use the same polynomial, turning comparisons into evaluations of small expressions.For codes of the same length and kernel dimension, equal reduced ranks are equivalent to equal ranks.Source: §4.2 · pp. 20–21
  • A nonlinear Z8-linear code with a = 1 matches a Z2Z4-linear code on length, kernel dimension, and rank.The parameters correspond as H̄_{1,b,c} and H_{b,c+3}.Source: §4.2 · p. 22

4.3 Collisions inside the family

  • The only distinct types sharing length, rank, and kernel dimension are (H^{1,2,t-6}, H^{2,0,t-5}) for t ≥ 7 and (H^{2,2,t-9}, H^{3,0,t-8}) for t ≥ 10.Each pair has equal length, rank, and kernel dimension.Source: §4.3 · p. 23
  • The only within-family collisions are (H1,2,t−6, H2,0,t−5) for t ≥ 7 and (H2,2,t−9, H3,0,t−8) for t ≥ 10; all other distinct types of the same length differ in rank or kernel dimension.The two pairs have respective common (rank, kernel dimension) values (t + 6, t − 3) and (t + 17, t − 5).Source: §4.3 · pp. 23–24

4.4 The number of classes separated by the two invariants

  • The number of distinct (rank, kernel dimension) pairs is A_t for 3 ≤ t ≤ 6, A_t−1 for 7 ≤ t ≤ 9, and A_t−2 for t ≥ 10; the pair classifies all types exactly when 3 ≤ t ≤ 6.Here A_3, A_4, A_5, and A_6 are 1, 1, 2, and 3, respectively.Source: §4.4 · p. 24

4.5 Comparison with the Z4-linear and the Z2Z4-linear Hadamard codes

  • The sole same-length, same-invariant match with a Z4- or Z2Z4-linear code is H2,1,t−7 and H5,t−9 for t ≥ 10, with rank t + 11 and kernel dimension t − 4.These codes have common length 2^t.Source: §4.5 · p. 26

4.6 Comparison with the Z8-linear Hadamard codes

  • Five cases have equal length, rank, and kernel dimension, with (rank, kernel dimension) pairs (t+3, t−2), (t+11, t−4), (t+21, t−6), (t+51, t−9), and (t+117, t−12) for t ≥ 5, 12, 13, 18, and 22, respectively.Source: §4.6 · p. 29
  • Only three tested table cells yield a nonnegative integer candidate m; the p = 2 case has no solution.The search covers 1 ≤ n ≤ p − 1 ≤ 6.Source: Table 2 · p. 30, §4.6 · p. 30
  • Of five Z8-linear coincidence families, four are new; the first, H^{1,1,t-4} and overline{H^{2,0,t-5}}, was previously observed for 5 ≤ t ≤ 11.The other four families were not previously identified.Source: §4.6 · p. 30
  • The last three Z8-linear coincidence families are invisible below t = 13, t = 18, and t = 22, respectively.Source: §4.6 · p. 30
  • Across the three comparisons, two infinite pair sequences remain unresolved within the family, one against Z2Z4-linear codes, and five against Z8-linear codes.Separating these pairs requires an invariant finer than rank and kernel dimension.Source: §4.6 · p. 30
  • For Z2^s alphabets with s ≥ 4, rank comparison is unavailable because no rank formula is known.Source: §4.6 · p. 30

5 Conclusions and further research

The paper gives a rank formula for the recursively constructed Z2Z4Z8-linear Hadamard codes and uses it with kernel dimension to compare code types. The conclusions also delimit the scope of those comparisons and identify open classification questions.

5 Conclusions and further research

  • The rank calculation combines coordinate functions from the Z2 and Z4 parts with a further carry space contributed by the top binary digit in the Z8 part.Modulo the first space, that top digit depends on the coordinate through its odd-entry positions and a linear form in the lowest message digits; the computed values agree with Magma results in [33, Tables 2–4].Source: §5 · pp. 30–31
  • The rank formula and kernel dimension leave exactly two pairs of distinct types of the same length sharing both invariants.The number of distinct (r, k) pairs is given by Corollary 4.13.Source: §5 · p. 31
  • The comparisons with Z4-, Z2Z4-, and Z8-linear Hadamard codes yield one coincidence family with the first two families and five with the Z8-linear family.Across these comparisons, the ranks are values of the same polynomial Θ at different parameter choices, reducing each comparison to a short Diophantine problem.Source: §5 · p. 31
  • The results concern the recursively constructed family and constructions proved permutation equivalent to it, not every code of the same abstract type.A generator matrix of type (4, 6, 12; 2, 0, 1) has a Gray image of rank 13, compared with rank 12 for H2,0,1.Source: §5 · p. 31
  • For lengths 2^t with 5≤t≤11, computer equivalence tests separated some pairs left unseparated by rank and kernel dimension, so those invariants do not complete classification.A complete classification requires a finer invariant, ideally one evaluable without a rank formula so comparison can extend to Z2s alphabets with s≥4.Source: §5 · p. 31
  • Further questions include extending rank results to alphabets beyond Z8 and explaining structurally the five Z8-linear coincidence families.The existence of Z4Z8-additive Hadamard codes with α1=0, α2≠0, and α3≠0 also remains open.Source: §5 · p. 31
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