Further results on binary convolutional codes with an optimum distance profile (Corresp.)

In a previous correspondence,’ a decoding procedurewhich uses continued fractions and which is applicable to a wideclass of algebraic codes including Goppa codes was presented. Theefficiency of this method is significantly increased. The efficiency of the decoding method in the above corre-spondencel can be significantly increased. In the following, allreferences will be to the equations and bibliography of the orig-inal paper.lIn the decoding process, the test (18) with the associatedmultiplications is not required; instead, the test for the correctconvergent can be taken asdeg (ri+l(x)) < k + deg (qi(x)),and therefore no multiplication is needed.This is shown as follows. From [6], we have(18’) qi-l(xh+l(x) + qib)ri(x) = 4~)pi-l(x)ri+l(x) + pi(x)r = u(x).Multiplying the above two equations by pi(x) and qi(x), re-spectively, and subtracting the first from the second yields Si(XMX) Pibhb)= qi(xh-l(x)r;+l(x)+ qi(x)pi(x)ri(x)Pi(x)qi-l(x)ri+l(x)-Pi(x)qi(x)ri(x)= ri+l(x)(qi(x)Pi-l(x)Pi(X)qi-l(X))= ri+l(x)(-1)‘.Since for the correct convergentU(X) = U(X) m(n)pi(x)/qi(x) ad deg (U(X)) < k,the rule (18’) follows. Also, U(X) can be obtained by means of u(x)= (-l)iri+i(x)/q;(x) when (18’) is satisfied. This is a particularlyattractive method of obtaining u(x) for low-rate codes.As a result of the new test (18’), it is seen that this procedureis of equivalent complexity to that given in [4].It can also be easily seen that the method presented in theabove correspondence’ can be used for decoding Goppa andBose-Chandhuri-Hocquenghem (BCH) codes using the standardsyndrome S(z) as developed in [9]. That is, if S(z) = q(z)/cr(z)modg(z), then, if we set S’(X) = xrS(l/x) where r = deg S(z); wecan use the syndrome S’(X) in the decoding method in the abovecorrespondencei. It will be noticed that this method, like theBerlekamp-Massey algorithm, starts with the “end” of thesyndrome while the method of [4] starts in the “middle” of thesyndrome.Two corrections should be made in the example in the abovecorrespondencel; namely, the remainders rz(x) and rg(x) shouldread r-z(x) = fi2x6 + p2x5 + x4 + xs + fi43c2+ /36x + fi andr&)= p5x5 + p2x4 + p4x3 + p3xz + /3% + 1, respectively. Manuscript received February 28,1977; revised June 1,1977.The author is at P.O. Box 645, Eatontown, NJ 07724.i D. M. Mandelbaum, IEEE Trans. Inform Theory, vol. IT-23, pp. 137-140, Jan.1977. 001%9448/78/0300-0268$00.75