Extending self-orthogonal codes

Extending self-orthogonal codes

In this short note we give an exact count for the number of self-dual codes over a finite field $F_q$ of oddcharacteristic containing a given self-orthogonal code. This generalizes an analogous result of MacWilliams, Sloane, andThompson over the field $F_2$ to arbitrary odd finite fields $F_q$.

___

  • [1] Artin E. Geometric Algebra. New York, NY, USA: Interscience Publishers Inc., 1957.
  • [2] Bassa A, Stichtenoth H. Self-dual codes better than the Gilbert–Varshamov bound. Designs, Codes and Cryptography 2019; 87: 173-182.
  • [3] MacWilliams FJ, Sloane NJA, Thompson JG. Good self dual codes exist. Discrete Mathematics 1972; 3: 153-162.
  • [4] Pless V. The number of isotropic subspaces in a finite geometry. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali 1965; 39: 418-421.
  • [5] Pless V, Pierce JN. Self-dual codes over GF(q) satisfy a modified Varshamov-Gilbert bound. Information and Control 1973; 23: 35-40.
  • [6] Scharlau W. Quadratic and Hermitian Forms. Berlin, Germany: Springer-Verlag, 1985.
  • [7] Segre B. Le geometrie di Galois. Annali di Matematica Pura ed Applicata 1959; 48 (4): 1-96 (in Italian).