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On Almost MDS codes holding $(n-k-1)$-designs and an open problem on Witt design

Huakai Wei, Shiyuan Qiang, Shaofang Hong

arXiv:2610.29744v1cs.ITmath.ITmath.NT

TL;DR

The paper addresses whether a dual-code 3-design proposed by Wang, Tang, and Ding is isomorphic to the complementary Witt spherical geometry design. It develops a framework for AMDS codes and their duals, and proves that the designs are isomorphic.

  • Problem

    Wang, Tang, and Ding asked whether the 3-design supported by minimum-weight words of their proposed cyclic dual code is isomorphic to the complementary Witt spherical geometry design.

    Source: §1 · p. 3

  • Method

    The paper characterizes the dual parameters and supported designs of AMDS codes whose minimum-weight words support the specified design, and derives formulas for dual subset-code weight enumerators.

    Source: §1 · p. 2

  • Results

    The dual-code minimum-weight 3-design is isomorphic to the complementary Witt spherical geometry design.

    Source: §1 · pp. 3–4

  • Takeaways & Limitations

    For dual subsets of length at least k+1, the weight enumerator depends on subset size, dimension, and λ, not on the particular subset.

    Source: §1 · p. 2

    The result is stated for AMDS codes with parameters [n, k, n−k] whose minimum-weight codewords support the specified design.

    Source: §3 · p. 11

Abstract

from arXiv · show

Let $\mathcal{C}$ be an almost maximum distance separable (AMDS) code over the finite fields with parameters $[n,k,n-k]$ whose minimum-weight codewords support an $(n-k-1)$-$(n,n-k,\lambda)$ design. We first determine the parameters of the dual code $\mathcal{C}^{\bot}$. We also characterize the design and its complementary design supported by the minimum weight codewords of $\mathcal{C}^{\bot}$. In particular, we present explicit formulas for the weight enumerators of all subset codes of $\mathcal{C}^{\bot}$ with length at least $k+1$. These formulas are completely determined by the cardinality of the subset, the dimension, together with the parameter $\lambda$, and are independent of the specific choice of subsets. As an application, we answer an open problem posed by Wang, Tang and Ding in 2023 by proving that the $3$-design supported by the minimum-weight codewords in their proposed cyclic dual code is isomorphic to the complementary design of the Witt spherical geometry design.

1. Introduction

The paper gives a unified account of duals of AMDS codes whose minimum-weight supports form specified designs, deriving dual parameters, design structure, and subset-code weight enumerators. It applies this framework to BCH codes and resolves Wang, Tang, and Ding’s open question about a dual 3-design and the complementary Witt spherical geometry design.

Theorem 1.1

  • For an AMDS code with parameters [n, k, n−k], the dual has parameters [n, n−k, k+1−λ], and its minimum-weight supports have a characterized complementary design.The paper also gives subset-code weight enumerators for subsets of size at least k+1, determined by subset size, dimension, and λ rather than subset identity.Source: §1 · p. 2

Motivation

  • Wang, Tang, and Ding studied cyclic codes whose minimum-weight words support 3-designs and posed whether a dual-code design is the complementary Witt design.Earlier work established families of codes related to Witt spherical geometry designs,,,.Source: §1 · pp. 2–3

Theorem 1.2

  • When pt > 2, the BCH code is AMDS with parameters [q + 1, q−3, 4], and its minimum-weight codewords support a 3-(q + 1, 4, pt−2) design.This is the third case of Theorem 1.2, under the stated congruence condition and t = gcd(m, m′).Source: §1 · p. 3

Theorem 1.3

  • The stated dual BCH cases include MDS parameters [q + 1, 3, q−1] and [q + 1, 4, q−2], and a further case with parameters [q + 1, 4, q−pt].The latter case supports the 3-design identified with the complementary Witt spherical geometry design.Source: §1 · p. 3
  • The minimum-weight dual codewords support a 3-design isomorphic to the complementary Witt spherical geometry design, answering Wang, Tang, and Ding’s open problem.The stated Witt design has parameters 3-(q + 1, pt + 1, 1).Source: §1 · pp. 3–4

2. Preliminary lemmas

The preliminary results supply the design-complement, linear-algebra, and finite-field tools used in the main proofs. For the BCH-code application, they connect a determinant condition on quadruples to projective transformations.

Design lemma

  • The complementary-design lemma converts a t-(v, k, λ) design into a design whose blocks are complements of the original blocks.The resulting block size is v − k; the lemma supplies the conversion used in the main design arguments.Source: §2 · p. 4

Linear-algebra lemmas

  • Lemma 2.4 decomposes coordinates by spans of parity-check columns and characterizes the dual subset code on the complementary coordinates.Its proof constructs independent columns and shows that coordinates in each resulting class are either all zero or all nonzero.Source: §2 · pp. 4–5

Finite-field and projective lemmas

  • The BCH lemmas establish that the determinant condition is preserved through equivalent finite-field formulations and relate the code’s minimum distance to gcd(2h+1, q+1).The stated criterion is d ≥ 4 exactly when gcd(2h+1, q+1) = 1.Source: §2 · pp. 6, 9
  • The determinant condition for four distinct elements of Uq+1 is equivalent to expressing the quadruple as the image of (0, ∞, 1, ξ) under a projective linear transformation.The parameter ξ lies in F*_{p^t} \ {1}, with t = gcd(m, m′).Source: §2 · p. 9

3. Proof of Theorem 1.1

The proof determines the dual code’s parameters and minimum-weight support design from the AMDS code’s design. It then uses the complementary design to obtain weight enumerators for dual subset codes.

Theorem 1.1

  • The proof identifies minimum-weight dual supports through parity-check-column spans, then uses those supports to construct the complementary design.For each set I of n − k − 1 coordinates, the proof considers indices whose columns lie in the span of the columns indexed by I.Source: §3 · pp. 11–12
  • The dual code has parameters [n, n − k, k + 1 − λ], and its minimum-weight supports form an (n − k − 1)-(n, k + 1 − λ, λ⊥) design.The complementary design is an (n − k − 1)-(n, n − k − 1 + λ, 1) design uniquely determined by the original design.Source: §3 · pp. 11–12
  • For subsets S of size k + r, the weight enumerator of C⊥(S) is obtained from the complementary design induced on S.The induced structure is an (|S| − k − 1)-(|S|, |S| − k − 1 + λ, 1) design.Source: §3 · pp. 13–14

4. Proofs of Theorems 1.2 and 1.3

The proofs classify the BCH code parameters across congruence cases and apply the dual-design result to the AMDS case. The resulting dual design is identified with a complementary Witt spherical geometry design.

Theorem 1.2

  • When m′ ≡ 0 (mod m), the BCH code is MDS with parameters [q + 1, q − 2, 4].The proof obtains dimension q − 2 from the degree of the generator-polynomial factors.Source: §4 · p. 14
  • When m′ ≠ 0 (mod m) and pt = 2, code C has dimension q − 3 and minimum distance 5, hence is MDS with parameters [q + 1, q − 3, 5].The proof establishes d > 4 and, by Lemma 2.5, d ≤ 5.Source: §4 · pp. 14–15
  • For pt > 2, C is an AMDS code with parameters [q + 1, q − 3, 4], and its minimum-weight codeword supports form a 3-(q + 1, 4, pt − 2) design.Source: §4 · p. 15

Theorem 1.3

  • The dual of the AMDS BCH code has parameters [q + 1, 4, q − pt], and its minimum-weight supports form a 3-(q + 1, q − pt, λ⊥) design.The complementary design is identified as a Witt spherical geometry design with parameters 3-(q + 1, p^gcd(m,i) + 1, 1).Source: §4 · p. 15
  • The complementary dual-design blocks are formed from projective images indexed by F_pt ∪ {∞}, yielding the Witt spherical geometry identification.The projective map π is a bijection between Fq ∪ {∞} and Uq+1.Source: §4 · p. 15

5. Final Remarks

The final remarks use Lemma 2.4 to establish NMDS-code criteria and derive applications of Theorem 1.1, including dual weight-enumerator results.

5. Final Remarks

  • Lemma 2.4 generalizes Proposition 14 in by yielding a unique minimum-weight dual codeword, up to scalar, whose support is disjoint from a given minimum-weight codeword.Source: §5 · p. 16
  • For an NMDS code, the dual is NMDS and every subset of size n−k+1 has a one-dimensional subset code; conversely, this condition implies the AMDS code is NMDS.The remarks also use Lemma 2.4 to give an alternative proof of Theorem 1.1(ii) in.Source: §5 · p. 16
  • Theorem 1.1 gives the dual weight enumerator for the AMDS codes studied in, matching the result in Example 28 there.Source: §5 · p. 16
  • For the case treated by Xiang, Tang and Liu, Theorem 1.1 yields the dual weight enumerator directly, without complex calculations.The result agrees with Example 2 in.Source: §5 · p. 17
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