Difference-set cyclic (DSC) code

Difference-set cyclic (DSC) code [1]

Description

Cyclic LDPC code constructed deterministically from a difference set. Certain DSC codes satisfy more redundant constraints than Gallager codes and thus can outperform them [2].

Notes

See book [3; Ch. 6] for a general theory of linear codes made from difference sets.

Cousins

\([2^m,m,2^{m-1}]\) Hadamard code— Hadamard difference sets are difference sets constructed from Hadamard matrices [3; Ch. 6].
Hyperoval code— Hyperoval difference sets yield DSC codes [4][3; Ch. 6].
Generalized RM (GRM) code— DSC codes can be subfield subcodes of GRM codes, and vice versa [3; Thm. 6.14].
Incidence-matrix projective code— Planar difference-set cyclic codes arise from the incidence vectors of the lines of a projective plane \(PG(2,p^s)\) [5; pgs. 397-398].

Member of code lists

Primary Hierarchy

Classical Domain

Binary Kingdom

Binary codeECC

Binary group-orbit codeECC

Linear binary codeLinear \(q\)-ary Linear code over \(\mathbb{Z}_q\) ECC

Low-density parity-check (LDPC) code\(q\)-ary LDPC Tanner Linear \(q\)-ary LRC Distributed-storage ECC

Multi-edge LDPC code

Protograph LDPC code\(q\)-ary LDPC Tanner Linear \(q\)-ary LRC Distributed-storage ECC

Quasi-cyclic LDPC (QC-LDPC) codeECC

Parents

Quasi-cyclic LDPC (QC-LDPC) code

Algebraic LDPC code\(q\)-ary LDPC Tanner Linear \(q\)-ary Linear code over \(\mathbb{Z}_q\) LRC Distributed-storage ECC

Cyclic linear binary codeLinear \(q\)-ary Linear code over \(\mathbb{Z}_q\) Cyclic Constacyclic Skew-cyclic ECC

Difference-set cyclic (DSC) code

Children

\([7,3,4]\) simplex code

The \([7,3,4]\) simplex code is the smallest difference-set cyclic code, arising from the lines of the projective plane \(PG(2,2)\) [5; pg. 397].

References

[1]
E. J. Weldon Jr., “Difference-Set Cyclic Codes”, Bell System Technical Journal 45, 1045 (1966) DOI
[2]
D. J. C. MacKay and M. C. Davey, “Evaluation of Gallager Codes for Short Block Length and High Rate Applications”, The IMA Volumes in Mathematics and its Applications 113 (2001) DOI
[3]
C. Ding, Codes from Difference Sets (WORLD SCIENTIFIC, 2014) DOI
[4]
A. Maschietti, “Difference Sets and Hyperovals”, Designs, Codes and Cryptography 14, 89 (1998) DOI
[5]
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes (Elsevier, 1977)

Page edit log

Victor V. Albert (2026-06-08) — most recent
Victor V. Albert (2023-05-04)

Cite as:

“Difference-set cyclic (DSC) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/difference_set, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/bits/tanner/qc/difference_set.yml.