Optimal lengths found for binary codes with self-orthogonality
Optimal self-orthogonal binary linear codes
Discrete MathematicsInformation Theory
Summary
The paper finds the shortest lengths possible for certain kinds of binary codes that are self-orthogonal and have specific minimum distances. The authors completely determine these lengths for codes with dimension eight and provide bounds for dimension nine. They also simplify and redo previous classifications for dimensions up to seven without using computers. As a result, they show some codes that cannot exist and reveal special divisibility properties of some optimal codes.
What this means in practice
- •For error correction engineers: Use exact shortest code lengths for reliable data transmission with self-orthogonal binary codes to optimize storage and error detection.
- •For cryptographic system developers: Incorporate guaranteed bounds on self-orthogonal code parameters to design more secure encoding schemes in cryptography.
Authors
Sascha Kurz
Abstract
We complete the determination of the minimum possible length of eight-dimensional binary self-orthogonal codes with given minimum distance. For dimension nine we give exact values and bounds for minimum distance at most $64$. By exploiting the geometric reformulation of linear codes as multisets of points, we fully redo the classification for dimension at most seven in a simplified and computer-free manner. As a by-product we prove the non-existence of $[93,9,44]_2$ codes and show several divisibility results for putative optimal nine-dimensional binary codes.