A linear feedback shift register, or LFSR, can implement an event counter by shifting whenever an event occurs. A single two-input exclusive-OR gate is often the only additional hardware necessary to allow a shift register to generate, by successive shifts, all of its possible nonzero values. The counting application requires that the number of shifts be recoverable from the LFSR contents so that further processing and analysis may be done. Recovering this number from the shift register value corresponds to a problem from number theory and cryptography known as the discrete logarithm. For some sizes of shift register, the maximal-length LFSR implementation requires more than a single gate, and for some, the discrete logarithm calculation is hard. The paper proposes for such size the use of certain one-gate LFSR's whose sequence lengths are nearly maximal, and which support easy discrete logarithms. These LFSR's have a concise mathematical characterization, and are quite common. The paper concludes by describing an application of these ideas in a computer hardware monitor, and by presenting a table that describes efficient LFSR's of size up to 64 bits.>

Some concatenated codes of length 128 and less are constructed. Nineteen of these codes are superior to the best previously known linear codes, as shown by the fact that the wellknown lower bound on the minimum distance of the concatenated code as the product of the minimum distances of its component codes exceeds the minimum distance of the best previously known code.

Slowly fading channels, such as a troposcatter communication link, are characterized by multiple error bursts which may last for many thousand channel digits, as well as by random errors. It is shown that even the best random error-correcting codes cannot be effective on this type of channel unless some time delay between code letters is introduced to reduce the correlation between channel errors within the code. In addition to the conventional approach of interleaving code letters to obtain error independence, two alternative coding approaches are considered which correct random errors as well as detect and correct multiple bursts, i.e., multiple channel fades. A concatenated coding approach, based on interleaving inner code blocks but not interleaving digits within an inner block, and a new multiple-burst correction code based on generalized burst trapping codes, are shown to represent attractive alternatives to the conventional interleaver approach.

If the period of anm-sequence is2^t - 1 = n_1 n_2withn_1relative prime ton_2, then the vectorm, a period of them-sequence, can be decomposed in a special way into two subsequences of lengthsn_1andn_2. The properties and structures of them-sequence and its subsequences are studied under such decomposition. The following applications are direct consequences. The correlation technique for the detection ofm- sequences can be improved. The weight distributions of a class of cyclic codes can be easily obtained. A scheme for improving the feedback communication is developed. And a method for designing cyclic codes with desirable weights is suggested.