New constructions for moderate-density parity-check (MDPC) codes using finite geometry are proposed. We design a parity-check matrix for the main family of binary codes as the concatenation of two matrices: the incidence matrix between points and lines of the Desarguesian projective plane and the incidence matrix between points and ovals of a projective bundle. A projective bundle is a special collection of ovals which pairwise meet in a unique point. We determine the minimum distance and the dimension of these codes, and we show that they have a natural quasi-cyclic structure. We consider alternative constructions based on an incidence matrix of a Desarguesian projective plane and compare their error-correction performance with regards to a modification of Gallager’s bit-flipping decoding algorithm. In this setting, our codes have the best possible error-correction performance after one round of bit-flipping decoding given the parameters of the code’s parity-check matrix.
|Article number||Special Issue: On Coding Theory and Combinatorics: In Memory of Vera Pless|
|Number of pages||24|
|Journal||Designs, Codes and Cryptography|
|Publication status||Published - 24 May 2022|
Bibliographical noteFunding Information:
We would like to thank the referees for their valuable comments. In particular, the code discussed in Sect. is due to one of their suggestions. The work of A. Neri was supported by the Swiss National Science Foundation through Grant No. 187711. The work of J. Rosenthal was supported by the Swiss National Science Foundation through Grant No. 188430.
© 2022, The Author(s).