A binary maximal length sequence (m-sequence) of period L = 2m- 1 can be generated by a binary m-stage linear feedback shift-register (LFSR). Tap connections of the LFSR corresponds to a binary primitive polynomial of degree m. m-sequences enjoy several well-known and unique properties. A new property for m-sequences, called cross-recurrence property, is presented in this paper and its potential applications are briefly outlined.

Two schemes for differential encoding of blockcoded M-ary PSK signals are presented and compared. Both proposed schemes perform differential encoding after channel decoding for minimizing the performance penalty associated with differential schemes. The first scheme requires the block codes to be rotationally invariant and linear, whereas the second requires rotational invariance only. Examples of bit error rate performance are also presented.>

The general concept of closest coset decoding (CCD) is presented, and a soft-decoding technique for block codes that is based on partitioning a code into a subcode and its cosets is described. The computational complexity of the CCD algorithm is significantly less than that required if a maximum-likelihood detector (MLD) is used. A set-partitioning procedure and details of the CCD algorithm for soft decoding of mod u mod u+v mod codes are presented. Upper bounds on the bit-error-rate (BER) performance of the proposed algorithm are combined, and numerical results and computer simulation tests for the BER performance of second-order Reed-Muller codes of length 16 and 32 are presented. The algorithm is a suboptimum decoding scheme and, in the range of signal-to-noise-power-density ratios of interest, its BER performance is only a few tenths of a dB inferior to the performance of the MLD for the codes examined.>

It is shown that the set of cycles of a linear feedback shift register with characteristic polynomial (1 + x)nare at most doubly adjacent.

Upper bounds on the worst case autocorrelation and the worst case cross correlation of linear PN sequences of the same period are obtained. The value of the worst case correlation curve at some specific points is found and using interpolation methods a smooth curve fitting to the correlation curve is formulated.

The cycle structure of a binary linear shift register with connection polynomialG(x)=(1+x)^{2}g(x), whereg(x)is a primitive polynomial of degreem-2over GF(2), is used to give several construction techniques for generation of shift-register sequences of lengthl=2^{m}-4. It is shown that a class of nonlinear deBruijn cycles, where the number of elements is proportional to2^{5m}, can be constructed. The obtained cycles can be generated by simplem-stage nonlinear feedback shift registers.

The minimum distance growth rate of unmerged codewords in a convolutional code is shown to depend upon the minimum average weight per branchw_{0}in the encoder state diagram. An upper bound onw_{0}is obtained for a large class of rate1/2codes which includes many of the best known classes of rate1/2codes. The hound is shown to be tight for short constraint length codes. A class of codes is defined to be asymptotically catastrophic ifw_{0}approaches zero for large constraint lengths. Several classes of rate1/2codes are shown to be asymptotically catastrophic. These include classes containing codes known to have large free distance. It is argued that the free distance alone is not a sufficient criterion to determine a codes performance with either Viterbi or sequential decoding. A code with a low distance growth rate will yield a high bit error probability and will not perform well with truncated Viterbi decoding.

Using the Euclidean Algorithm for polynomials over GF(q), an algebraic technique for the generation of q-ary shift register sequences of arbitrary length l, 1 ≤ l ≤ qm, is obtained, where q is a power of a prime number, q = pn, and m is the number of shift register stages.

An upper bound on the bit error probability due to truncation of the path length in Viterbi decoding is obtained for any given convolutional code. This bound is then used to determine the path length at which the additional error probability due to truncation becomes negligible compared to the maximum likelihood decoding error probability. These results are tested by simulation using several short constraint length codes.