[Paper Review] Infinite families of $2$-designs from two classes of binary cyclic codes with three nonzeros

Xiaoni Du, Rong Wang+2 more|arXiv (Cornell University)|Mar 19, 2019

graph theory and CDMA systems6 references7 citations

TL;DR

This paper constructs infinite families of 2-designs from two classes of binary cyclic codes with three nonzeros: one related to triple-error-correcting BCH codes and the other to generalized Kasami-case codes. By determining their weight distributions and proving affine-invariance, the authors explicitly derive the parameters of the 2-designs, with five 3-designs found in the dual of the BCH code when m=4.

ABSTRACT

Combinatorial $t$-designs have been an interesting topic in combinatorics for decades. It is a basic fact that the codewords of a fixed weight in a code may hold a $t$-design. Till now only a small amount of work on constructing $t$-designs from codes has been done. In this paper, we determine the weight distributions of two classes of cyclic codes: one related to the triple-error correcting binary BCH codes, and the other related to the cyclic codes with parameters satisfying the generalized Kasami case, respectively. We then obtain infinite families of $2$-designs from these codes by proving that they are both affine-invariant codes, and explicitly determine their parameters. In particular, the codes derived from the dual of binary BCH codes hold five $3$-designs when $m=4$.

Motivation & Objective

• To determine the weight distributions of two classes of binary cyclic codes with three nonzeros: one from triple-error-correcting BCH codes and the other from generalized Kasami-case codes.
• To prove that these codes are affine-invariant, enabling the construction of t-designs from their codewords.
• To explicitly compute the parameters of the 2-designs derived from the extended dual codes of these cyclic codes.
• To investigate the existence and structure of 3-designs in the extended dual of the BCH code when m=4.
• To extend the known constructions of t-designs from linear codes by leveraging cyclic codes with three nonzeros and their weight distributions.

Proposed method

• Derived the weight distribution of the first class of codes using Gauss sums and character sum techniques, specifically analyzing the exponential sum $ S(a,b,c) = rac{1}{2} imes ext{Tr}_1^m(aeta^5 + beta^3 + ceta) $ over $ eta eq 0 $ in $ ext{GF}(2^m) $.
• Used the Pless power moments identity to solve for the weight enumerator coefficients $ A_i $, yielding explicit formulas for all nonzero weights.
• Proved that the extended dual codes $ ar{ ext{C}_1^ot} $ and $ ar{ ext{C}_2^ot} $ are affine-invariant by verifying that their defining sets are closed under the componentwise partial order $ rianglelefteq $.
• Applied Theorem 3, which guarantees that affine-invariant codes yield 2-designs from the supports of codewords of any fixed nonzero weight.
• Computed the number of blocks in each 2-design using the relation $ b = rac{1}{inom{k}{2}} imes inom{v}{2} imes rac{A_k}{ ext{number of blocks per block type}} $, derived from the design identity.
• Verified the 3-designs in $ ar{ ext{C}_1^ot} $ at $ m=4 $ by checking that the supports of codewords of certain weights form 3-designs with $ inom{v}{3} imes ext{constant} $ blocks.

Experimental results

Research questions

• RQ1 Do the codewords of fixed weight in the dual of the triple-error-correcting binary BCH code with three nonzeros form a 2-design?
• RQ2 Can the weight distribution of a binary cyclic code with three nonzeros be fully determined using character sum methods?
• RQ3 Under what conditions does the extended dual of a binary cyclic code with three nonzeros yield a 2-design?
• RQ4 Are there 3-designs in the extended dual of the triple-error-correcting BCH code when $ m=4 $?
• RQ5 Can infinite families of 2-designs be systematically constructed from cyclic codes related to the generalized Kasami case?

Key findings

• The weight distribution of the first class of codes (related to triple-error-correcting BCH codes) was fully determined, yielding seven distinct nonzero weights: $ 2^{2s-1} $, $ 2^{2s-1} ackslashpm 2^{s-1} $, $ 2^{2s-1} ackslashpm 2^s $, and $ 2^{2s-1} ackslashpm 2^{s+1} $.
• The number of codewords of weight $ 2^{2s-1} $ is $ 29 imes 2^{6s-6} - 33 imes 2^{4s-6} + 17 imes 2^{2s-4} - 1 $.
• The number of codewords of weight $ 2^{2s-1} - 2^{s-1} $ is $ rac{1}{15}(3 imes 2^{6s} + 3 imes 2^{5s} + 5 imes 2^{4s} + 5 imes 2^{3s} - 2^{2s+3} - 2^{s+3}) $.
• The extended dual code $ ar{ ext{C}_1^ot} $ is affine-invariant, ensuring that the supports of codewords of any fixed nonzero weight form a 2-design.
• For $ m=4 $, the code $ ar{ ext{C}_1^ot} $ contains five distinct 3-designs, as confirmed by the weight distribution and design parameter verification.
• The second class of codes (generalized Kasami case) also yields infinite families of 2-designs via the same affine-invariance and weight distribution approach, with parameters explicitly computed.