2025/08/22 by Alshuhail, Altaf
#FOS: Mathematics #Group Theory (math.GR) #Primary 94 B05 #Rings and Algebras (math.RA) #Secondary 16 A10
paper · doi:10.48550/arxiv.2508.16344
We consider codes over the two semi-local non-unital rings of order six, H23 = ⟨ a,b | 2a=0, 3b = 0, a2=a, b2 = 0, ab = 0 = ba ⟩, and H32 = ⟨ a,b | 2a=0, 3b = 0, a2=0, b2 = b, ab = 0 = ba ⟩. with respect to a symplectic inner product. Via the decomposition C = a Ca⊕ b Cb, any Hz-code splits into a binary component Ca and a ternary component Cb; this yields characterizations of symplectic self-orthogonal, self-dual, and quasi self-dual (QSD) codes, and allows the introduction of symplectic nice and symplectic linear complementary dual (LCD) codes. Using the automorphism groups of Ca and Cb and double-coset enumeration, we classify, up to permutation equivalence, all symplectic self-orthogonal and QSD Hz-codes of short even lengths.