The theory of error-correcting codes is a key pillar in modern digital communications. By introducing redundancy into codewords, error-correcting codes make messages robust against noises in communication channels. In 1948, Shannon proved that arbitrarily reliable communications are possible with the help of error-correcting codes. Thereafter, researchers have been exploring proper codes with efficient encoding and decoding algorithms.

BCH codes are a family of cyclic codes with guaranteed error-correcting capabilities and efficient decoding algorithms. They are employed in a wide range of applications from satellite communications to solid-state drives. However, their dimensions and minimum distances are seldom settled.

In this thesis, we investigate the parameters of BCH code...[ Read more ]

The theory of error-correcting codes is a key pillar in modern digital communications. By introducing redundancy into codewords, error-correcting codes make messages robust against noises in communication channels. In 1948, Shannon proved that arbitrarily reliable communications are possible with the help of error-correcting codes. Thereafter, researchers have been exploring proper codes with efficient encoding and decoding algorithms.

BCH codes are a family of cyclic codes with guaranteed error-correcting capabilities and efficient decoding algorithms. They are employed in a wide range of applications from satellite communications to solid-state drives. However, their dimensions and minimum distances are seldom settled.

In this thesis, we investigate the parameters of BCH codes. Dimensions of BCH codes with three different lengths are explored in a much larger range of defining sets than in the previous literature. For suitable cases, we determine the dimensions of BCH codes constructed from every feasible defining set within an interval; for other cases, we demonstrate the dimensions of those constructed from representative defining sets. For minimum distances of BCH codes, we summarized the classic conclusions and highlight new families of BCH codes whose true minimum distances can be determined by innovative methods. We also provide abundant examples of BCH codes with determined parameters, some of which are optimal.