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 /spl nu/-dimensional vector subspace S of GF (2/sup m/). SSRS codes are constructed using properties of the Galois field GF(2/sup m/). They are not linear over the field GF(2/sup /spl nu//), 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(2/sup /spl nu//) is a subfield of GF(2/sup m/), it may not be the best /spl nu/-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.

An explicit expression is given for the weights of any group code. The exponential weight of a vector, an element of a finite field, is introduced. It is a primitiven^{th}root of unity raised to the weight of a vector. Using the general representation of group codes via polynomials, a formula is obtained for the exponential weight of a code word as a function of the parameters of the general representation. The coefficients of the formula are obtained using elementary multiplication on the cyclic group of the nth roots of unity. The formula is, in general, quite complicated and requires a computer for computation of its explicit form. However, for the new practical codes of a highly algebraic nature (Bose-Chaudhuri, Cyclic, Polynomial, Jump-Shift Register codes which are characterized by simple field parameters and polynomials), the formula emerges more pliable. It is hoped that use of the formula will yield more precise error-correcting properties of a given group code (minimum weight of ail non-zero vectors of a code) and the distribution of weights of a code.