SKEW-INVOLUTORY MATRICES OVER FINITE FIELDS AND THEIR APPLICATIONS TO SELF-DUAL AND LCD CODES
Advances in Mathematics of Communications, cilt.25, ss.247-266, 2026 (SCI-Expanded, Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 25
- Basım Tarihi: 2026
- Doi Numarası: 10.3934/amc.2026051
- Dergi Adı: Advances in Mathematics of Communications
- Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Applied Science & Technology Source, MathSciNet, zbMATH
- Sayfa Sayıları: ss.247-266
- Anahtar Kelimeler: LCD codes, linear codes, self orthogonal codes, Skew involutory matrices
- Ankara Üniversitesi Adresli: Evet
Özet
What does it mean for a matrix to be its own negative inverse? This article explores a striking condition in linear algebra: an invertible matrix over a finite field whose inverse equals its negative. We call such matrices skew-involutory (SI). We prove that SI matrices exist in every dimension precisely when −1 is a square in Fq, and otherwise only in even dimensions. We characterize Skew Involutory Symmetric (SIS) matrices in small dimensions, derive explicit counting formulas for |SI(n)| in both cases where −1 is a square and where x2 + 1 is irreducible over Fq, and present two constructive methods (a Gröbner-basis approach and a companion-block method). We demonstrate their utility by producing explicit SIS matrices that yield self-dual and LCD codes with good parameters, including small-length examples that are simultaneously MDS and self-dual or LCD. Illustrative computational results, covering sizes up to n = 6 and fields up to q = 27, confirm the effectiveness of our approach and illustrate its potential for larger parameters whenever suitable SIS matrices are available.