[Paper Review] An infinite family of linear codes supporting 4-designs

Chunming Tang, Cunsheng Ding|arXiv (Cornell University)|Jan 1, 2020

Coding theory and cryptography20 references4 citations

TL;DR

This paper resolves a 71-year-old open problem by constructing an infinite family of BCH codes over $\mathrm{GF}(2^{2m+1})$ of length $2^{2m+1}+1$ that support an infinite family of $4$-designs with parameters $4$-$(2^{2m+1}+1, 6, 2^{2m}-4)$. The construction leverages properties of elementary symmetric polynomials and cyclic code structures to prove that minimum weight codewords in these codes yield $t$-designs, establishing the first known infinite family of linear codes supporting $4$-designs.

ABSTRACT

The first linear code supporting a $4$-design was the $[11, 6, 5]$ ternary Golay code discovered in 1949 by Golay. In the past 71 years, sporadic linear codes holding $4$-designs or $5$-designs were discovered and many infinite families of linear codes supporting $3$-designs were constructed. However, the question as to whether there is an infinite family of linear codes holding an infinite family of $t$-designs for $t\geq 4$ remains open for 71 years. This paper settles this long-standing problem by presenting an infinite family of BCH codes of length $2^{2m+1}+1$ over $\mathrm{GF}(2^{2m+1})$ holding an infinite family of $4$-$(2^{2m+1}+1, 6, 2^{2m}-4)$ designs. Moreover, an infinite family of linear codes holding the spherical design $S(3, 5, 4^m+1)$ is presented.

Motivation & Objective

• To resolve the long-standing open problem of whether an infinite family of linear codes exists that supports an infinite family of $t$-designs for $t \geq 4$.
• To construct explicit linear codes over finite fields that support $4$-designs, particularly focusing on near-MDS codes.
• To provide a coding-theoretic construction of spherical $3$-designs $S(3,5,4^m+1)$ via codewords of specific weights.
• To demonstrate that the Assmus-Mattson Theorem and automorphism group methods are insufficient to prove $3$-designs in certain cases, highlighting the need for alternative approaches.
• To explore the structural properties of linear codes that support $t$-designs and to identify conditions under which such designs arise from codeword supports.

Proposed method

• Constructing a family of BCH codes $\mathsf{C}_{(q,q+1,4,1)}$ of length $q+1 = 2^{2m+1}+1$ over $\mathrm{GF}(q)$ with $q = 2^{2m+1}$, using the defining set approach.
• Analyzing the weight distribution of the code and its dual using properties of elementary symmetric polynomials and trace functions over finite fields.
• Applying the Assmus-Mattson Theorem to identify conditions under which codewords of fixed weight support $t$-designs, particularly for $t=3$ and $t=4$.
• Establishing isomorphisms between incidence structures formed by codeword supports and known combinatorial designs, such as $S(3,5,4^m+1)$ and $4$-designs.
• Using Lemma 31 and Lemma 34 to prove that the supports of minimum weight codewords in $\mathsf{C}_{(q,q+1,4,1)}$ correspond to $5$-subsets and $6$-subsets forming $3$-designs.
• Verifying the design parameters via combinatorial identities and binomial coefficient computations, including the use of Equation (40) and Corollary 7 to derive $\lambda$-values for $3$-designs in the dual code.

Experimental results

Research questions

• RQ1 Does there exist an infinite family of linear codes that supports an infinite family of $t$-designs for $t \geq 4$?
• RQ2 Can a coding-theoretic construction yield a $4$-design using codewords of fixed weight in a linear code over a finite field?
• RQ3 Are the spherical $3$-designs $S(3,5,4^m+1)$ realizable as support designs of linear codes, and if so, via which code parameters?
• RQ4 Why do the Assmus-Mattson Theorem and automorphism group analysis fail to prove $3$-designs in certain cases, such as for $\mathsf{C}_{(q,q+1,4,1)}$?
• RQ5 Is there a structural difference between the spherical $3$-designs $S(3,5,4^m+1)$ constructed via group actions and those constructed via linear codes?

Key findings

• The paper constructs an infinite family of linear codes over $\mathrm{GF}(2^{2m+1})$ of length $2^{2m+1}+1$ that support $4$-$(2^{2m+1}+1, 6, 2^{2m}-4)$ designs, resolving a 71-year-old open problem.
• The minimum weight codewords in $\mathsf{C}_{(q,q+1,4,1)}$ with $q = 2^{2m+1}$ support a $3$-$(2^m+1, 5, 1)$ Steiner system $S(3,5,2^m+1)$, which is isomorphic to the spherical design $S(3,5,4^m+1)$ for $m \in \{2,3\}$.
• The codewords of weight $6$ in $\mathsf{C}_{(q,q+1,4,1)}$ support a $3$-$(q+1,6,\frac{(q-4)(q-16)}{6})$ design when $m \geq 6$, with $q = 2^m$.
• The dual code $\mathsf{C}_{(q,q+1,4,1)}^\perp$ supports a $3$-$(q+1, q-5, \lambda)$ design with $\lambda = \frac{(q-4)^2}{120}\binom{q-5}{3}$, confirming the existence of $3$-designs in the dual.
• The example with $q = 2^4 = 16$ shows a $[17,11,5]$ code whose minimum weight codewords support a $S(3,5,17)$ Steiner system, and the dual $[17,6,11]$ code supports a $3$-$(17,11,198)$ design.
• The paper demonstrates that the Assmus-Mattson Theorem cannot be used to prove $3$-designs in this case, indicating limitations of classical design-theoretic tools in this context.