This paper considers the optimal generator matrices of a given binary cyclic code over a binary symmetric channel with crossover probability p→0 when the goal is to minimize the probability of an information bit error. A given code has many encoder realizations and the information bit error probability is a function of this realization. Our goal here is to seek the optimal realization of encoding functions by taking advantage of the structure of the codes, and to derive the probability of information bit error when possible. We derive some sufficient conditions for a binary cyclic code to have systematic optimal generator matrices under bounded distance decoding and determine many cyclic codes with such properties. We also present some binary cyclic codes whose optimal generator matrices are non-systematic under complete decoding.

This paper considers reduction of the peak-to-average power ratio (PAPR) of M-quadrature amplitude modulation (QAM) signals in orthogonal frequency division multiplexing (OFDM) systems. It is known that a 16-QAM or 64-QAM constellation can be written as the vector sum of two or three QPSK constellations respectively. We can then use the Golay complementary sequences over Z4 to construct 16-QAM or 64-QAM OFDM sequences with low PAPR. In this paper, we further examine the squared Euclidean distance of these M-QAM sequences and their variations. Our goal here is to combine the block coded modulation (BCM) and Golay complementary sequences to trade off the PAPR, the code rate, and the squared Euclidean distance of M-QAM OFDM signals. In particular, some 16-QAM and 64-QAM OFDM sequences with low PAPR and large squared Euclidean distance are presented.