Subspace subcodes of Reed-Solomon codes

We introduce a class of nonlinear cyclic error-correcting codes, which we call subspace subcodes of Reed-Solomon (SSRS) codes. An SSRS code is a subset of a parent Reed-Solomon (RS) code consisting of the RS codewords whose components all lie in a fixed ν-dimensional vector subspace S of GF (2m). SSRS codes are constructed using properties of the Galois field GF(2m). They are not linear over the field GF(2ν), which does not come into play, but rather are Abelian group codes over S. However, they are linear over GF(2), and the symbol-wise cyclic shift of any codeword is also a codeword. Our main result is an explicit but complicated formula for the dimension of an SSRS code. It implies a simple lower bound, which gives the true value of the dimension for most, though not all, subspaces. We also prove several important duality properties. We present some numerical examples, which show, among other things, that (1) SSRS codes can have a higher dimension than comparable subfield subcodes of RS codes, so that even if GF(2ν) is a subfield of GF(2m), it may not be the best ν-dimensional subspace for constructing SSRS codes; and (2) many high-rate SSRS codes have a larger dimension than any previously known code with the same values of n, d, and q, including algebraic-geometry codes. These examples suggest that high-rate SSRS codes are promising candidates to replace Reed-Solomon codes in high-performance transmission and storage systems

Properties and Construction of Polar Codes

Recently, Ar{\i}kan introduced the method of channel polarization on which one can construct efficient capacity-achieving codes, called polar codes, for any binary discrete memoryless channel. In the thesis, we show that decoding algorithm of polar codes, called successive cancellation decoding, can be regarded as belief propagation decoding, which has been used for decoding of low-density parity-check codes, on a tree graph. On the basis of the observation, we show an efficient construction method of polar codes using density evolution, which has been used for evaluation of the error probability of belief propagation decoding on a tree graph. We further show that channel polarization phenomenon and polar codes can be generalized to non-binary discrete memoryless channels. Asymptotic performances of non-binary polar codes, which use non-binary matrices called the Reed-Solomon matrices, are better than asymptotic performances of the best explicitly known binary polar code. We also find that the Reed-Solomon matrices are considered to be natural generalization of the original binary channel polarization introduced by Ar{\i}kan.Comment: Master thesis. The supervisor is Toshiyuki Tanaka. 24 pages, 3 figure

Codes for protection from synchronization loss and additive errors

Codes for protection from synchronization loss and additive error