Characterization and constructions of binary self-orthogonal singly-even linear codes
Kangquan Li, Hao Chen, Wengang Jin, Longjiang Qu
TL;DR
This work advances binary self-orthogonal code theory by fully characterizing SO codes and, in particular, the singly-even case. It develops necessary and sufficient criteria for constructing SO singly-even codes from Boolean and vectorial Boolean functions, and provides multiple infinite families with minimal, few-weight, AB-violating properties. It also introduces general methods to generate many SO codes from existing SO codes and from known doubly-even codes, including two-weight and four-weight examples. The results have implications for quantum information, lattice design, secret sharing, and related combinatorial constructions by expanding the catalog of minimal AB-violating codes with explicit weight distributions.
Abstract
Recent research has focused extensively on constructing binary self-orthogonal (SO) linear codes due to their applications in quantum information theory, lattice design, and related areas. Despite significant activity, the fundamental characterization remains unchanged: binary SO codes are necessarily even (all codeword weights even), while doubly-even codes (weights divisible by 4) are automatically SO. This paper advances the theory by addressing the understudied case of singly-even (even but not doublyeven) SO codes. We first provide a complete characterization of binary SO linear codes, and a necessary and sufficient condition for binary SO singly-even linear codes is given. Moreover, we give a general approach to generating many binary SO linear codes from two known SO linear codes, yielding three infinite classes of binary SO singly-even linear codes with few weights. Note that these new codes are also minimal and violate the Aschikhmin-Barg condition. Their weight distributions are determined. Furthermore, we give a necessary and sufficient condition for a Boolean function f such that the linear code proposed from f via a well-known generic construction is SO singly-even, and a general approach to constructing Boolean functions satisfying this condition is provided, yielding several infinite classes of binary SO singly-even minimal linear codes with few weights. Finally, we would like to emphasize that using the methods in this paper, we can construct more binary linear codes that are SO, singly-even, minimal, violating the AB condition, and with few weights at the same time.