New multilevel codes over GF(q)

In this paper, we apply set partitioning to multi-dimensional signal spaces over GF(q), particularly GFq-l(q) and GFq(q), and show how to construct both multi-level block codes and multi-level trellis codes over GF(q). We present two classes of multi-level (n, k, d) block codes over GF(q) with block length n, number of information symbols k, and minimum ,1-1 • { d } distance d_n >_ d, where n = nln2,1c = n-_=o mm [i-_]-1,n2 , nl = q1 or q, n2 = q 1,q, or q -t1, and [x] is the smallest integer larger than or equal to x. These two classes of codes use Reed-Solomon codes as component codes. They can be easily decoded as block length q 1 Reed-Solomon codes or block length q or q + 1 extended Reed-Solomon codes using multi-stage decoding. Many of these codes have larger distances than comparable q-ary BCH codes. Longer block codes can be constructed by using q-ary BCH codes, or other q-ary block codes, as component codes. Low rate q-ary convolutional codes, word error-correcting convolutional codes, and binary-to-q-ary convolutional codes can also be used to construct multi-level trellis codes over GF(q) or binary-to-q-ary trellis codes, some of which have better performance than the above block codes. All of the new codes have simple decoding algorithms based on hard decision multi-stage decoding. • This work was supported by NASA Grant NAG5-557 and NSF Grant NCR89-03429. (NA_A-C_-186862} N_W MULTI-LEVeL C_OES JV_R N90-2559_ GF(q) (Notre Dame Univ.) 24 p CSCL 09_ Uncl ds G_/O1 0293539