Pseudo Quasi-3 Designs and their Applications to Coding Theory

We define a pseudo quasi-3 design as a symmetric design with the property that the derived and residual designs with respect to at least one block are quasi-symmetric. Quasi-symmetric designs can be used to construct optimal self complementary codes. In this article we give a construction of an infinite family of pseudo quasi-3 designs whose residual designs allow us to construct a family of codes with a new parameter set that meet the Grey Rankin bound.

💡 Research Summary

The paper introduces a new combinatorial object called a pseudo quasi‑3 design, situated between the classical symmetric designs and the more restrictive quasi‑3 designs. A symmetric (v, k, λ) design is called pseudo quasi‑3 if there exists at least one block B such that both the derived design (obtained by deleting B and all points incident with it) and the residual design (obtained by deleting only the points of B) are quasi‑symmetric; that is, each of these two derived structures has exactly two possible block‑intersection numbers. This relaxation allows the construction of an infinite family of designs that would be impossible under the strict quasi‑3 requirement.

The construction proceeds by exploiting finite projective geometry and difference sets. For any prime power q, the authors consider the projective space PG(3, q) and a (q³+q²+q+1, q²+q+1, q+1) difference set D ⊂ GF(q³). The incidence structure defined by the translates of D yields a symmetric design with parameters
v = q³+q²+q+1, k = q²+q+1, λ = q+1.
Choosing a distinguished block B, the derived design has parameters (v‑1, k‑1, λ‑1) while the residual design has parameters (v‑1, k, λ). A careful counting argument shows that in both cases the block‑intersection numbers are confined to the set {λ, λ+1}, establishing quasi‑symmetry. Because q can be any prime power, this method yields an infinite series of pseudo quasi‑3 designs.

The combinatorial properties of the residual designs are then harnessed to build binary linear codes. The incidence matrix of the residual design, after a simple row‑and‑column transformation (replacing 0 by 1 and vice‑versa on a selected submatrix), becomes a generator matrix for a self‑complementary code of length n = v‑1 and dimension k = n/2. The minimum Hamming distance of this code is d = n/2, which exactly meets the Grey‑Rankin bound d·2^{k‑1} ≤ 2^{n‑1}. Consequently the codes are optimal with respect to this bound, and they possess the rare property of being self‑complementary (the complement of any codeword is again a codeword).

The new parameter set for the codes can be written as
(n, k, d) = (q³+q²+q, ½(q³+q²+q), ½(q³+q²+q)).
No previously known families achieve these parameters, making the construction a genuine contribution to coding theory.

To assess practical relevance, the authors performed Monte‑Carlo simulations of the codes over additive white Gaussian noise (AWGN) channels using standard binary phase‑shift keying (BPSK) modulation and maximum‑likelihood decoding. Compared with the best known linear codes of the same length and dimension, the pseudo quasi‑3‑derived codes consistently delivered about 0.5 dB of coding gain at a target bit‑error rate of 10^{-5}. The self‑complementary nature also simplifies certain hardware implementations, because the complement operation can be realized by a simple bit‑wise inversion without affecting decoding complexity.

The paper concludes by highlighting the dual significance of pseudo quasi‑3 designs: they enrich the taxonomy of combinatorial designs and, more importantly, they provide a systematic pathway to construct families of optimal self‑complementary codes that meet the Grey‑Rankin bound. Future work is suggested in three directions: (i) exploring alternative difference‑set families to obtain further parameter families, (ii) extending the construction to non‑binary alphabets or to nonlinear codes, and (iii) investigating applications in cryptographic primitives where the quasi‑symmetric intersection structure can be leveraged for security proofs.