4 research outputs found
Algebraic constructions of LDPC codes with no short cycles
An algebraic group ring method for constructing codes with no short cycles in
the check matrix is derived. It is shown that the matrix of a group ring
element has no short cycles if and only if the collection of group differences
of this element has no repeats. When applied to elements in the group ring with
small support this gives a general method for constructing and analysing low
density parity check (LDPC) codes with no short cycles from group rings.
Examples of LDPC codes with no short cycles are constructed from group ring
elements and these are simulated and compared with known LDPC codes, including
those adopted for wireless standards.Comment: 13 pages, 8 figures in pdf forma
Coding Theory: the unit-derived methodology
The unit-derived method in coding theory is shown to be a unique optimal
scheme for constructing and analysing codes. In many cases efficient and
practical decoding methods are produced. Codes with efficient decoding
algorithms at maximal distances possible are derived from unit schemes. In
particular unit-derived codes from Vandermonde or Fourier matrices are
particularly commendable giving rise to mds codes of varying rates with
practical and efficient decoding algorithms. For a given rate and given error
correction capability, explicit codes with efficient error correcting
algorithms are designed to these specifications. An explicit constructive proof
with an efficient decoding algorithm is given for Shannon's theorem. For a
given finite field, codes are constructed which are `optimal' for this field
Convolutional codes from unit schemes
Convolutional codes are constructed, designed and analysed using row and/or
block structures of unit algebraic schemes. Infinite series of such codes and
of codes with specific properties are derived. Properties are shown
algebraically and algebraic decoding methods are derived. For a given rate and
given error-correction capability at each component, convolutional codes with
these specifications and with efficient decoding algorithms are constructed.
Explicit prototype examples are given but in general large lengths and large
error capability are achievable. Convolutional codes with efficient decoding
algorithms at or near the maximum free distances attainable for the parameters
are constructible. Unit memory convolutional codes of maximum possible free
distance are designed with practical algebraic decoding algorithms.
LDPC (low density parity check) convolutional codes with efficient decoding
schemes are constructed and analysed by the methods. Self-dual and
dual-containing convolutional codes may also be designed by the methods;
dual-containing codes enables the construction of quantum codes.Comment: This version has substantive changes from previous version
Algebraic constructions of LDPC codes with no short cycles.
An algebraic group ring method for constructing codes with no short cycles in the check matrix is derived. It is shown that the matrix of a group ring element has no short cycles if and only if the collection of group differences of this element has no repeats. When applied to elements in the group ring with small support this gives a general method for constructing and analysing low density parity check (LDPC) codes with no short cycles from group rings. Examples of LDPC codes with no short cycles are constructed from group ring elements and these are simulated and compared with known LDPC codes, including those adopted for wireless standards.