msequence
msequence A periodic sequence of symbols generated by a linear feedback shift register whose feedback coefficients form a primitive polynomial. A qary register (with q prime) whose generating polynomial is of degree n will have period q^{n} – 1, provided that the initial state is nonzero, and its contents will proceed through all the nonzero qary ntuples. The termwise moduloq sum of two msequences is another msequence: the msequences (of a given generating polynomial), together with the zero sequence, form a group.
The term is short for maximumlength sequence. It is so called because the generating shift register only has q^{n} states, and so such a register (with arbitrary feedback logic) cannot generate a sequence whose period exceeds q^{n}. But with linear logic the zero state must stand in a loop of its own (see Good–de Bruijn diagram) and so the period of a linear feedback register cannot exceed q^{n} – 1. This period, which can be achieved when and only when the polynomial is primitive, is therefore the maximum that can be achieved.
msequences have many useful properties. They are employed as pseudorandom sequences, errorcorrecting codes (as they stand, or shortened, or extended), and in determining the time response of linear channels (see convolution). See also simplex codes.
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