Ultimate linear block and convolutional codes
Ted Hurley
TL;DR
This work develops a unified unit-derived framework for constructing linear block and convolutional codes with tunable length, rate, distance, and code type. By exploiting unit schemes $UV=I_n$ built from orthogonal, Fourier/Vandermonde, Hadamard, and group-ring structures, the authors generate DC, LCD, self-dual, and quantum variants, as well as LDPC families, and extend these designs to memory-rich convolutional codes with explicit generator and control matrices. The approach enables systematic, algebraic design of broad code families, including MDDS and QECCs via CSS, and provides prototype examples (e.g., Hamming, Golay, Hadamard-based constructions) to illustrate the method and guide large-length scalable constructions. The methods offer practical utility through efficient decoding options (Viterbi, syndrome decoding) and the capacity to tailor codes to specific fields and characteristics, yielding versatile options for communications and quantum information processing with potential performance benefits, especially for LDPC and convolutional codecs.
Abstract
Codes considered as structures within unit schemes greatly extends the availability of linear block and convolutional codes and allows the construction of these codes to required length, rate, distance and type. Properties of a code emanate from properties of the unit from which it was derived. Orthogonal units, units in group rings, Fourier/Vandermonde units and related units are used to construct and analyse linear block and convolutional codes and to construct these to predefined length, rate, distance and type. Self-dual, dual containing, quantum error-correcting and linear complementary dual codes are constructed for both linear block and convolutional codes. Low density parity check linear block and convolutional codes are constructed with no short cycles in the control matrix.