Optimum linear coding in continuous-time communication systems with noisy side information at the decoder (Corresp.)

interval is taken as T s. A similar problem, but in discrete time, more, if there is an m such that Tableh(m)#Table',(Y) for every has been considered in [l] where the authors have proposed an ad hoc linear encoding-decoding scheme which does not Y, then II E,H and Table~(k)=Table~(Z). With the computer asymptotically achieve the known performance bound for the we found the tables and from them we found that &,H)= 1. case when the side information is transmitted noise-free. In a This was done finding an m6 for which Tabler(k)#TableF(l) whenever Tablei(k)=Tablei(l). In this way we found 8(H) and recent note [2] they have improved on this scheme by consider-concluded that 19(G,') = 0(HI') and therefore Gp = HF. There is ing the case when the transmitter can also use the side information , and have shown within this context the optimal linear one exception: Hf (the group for X4+X+ 1) has an extra scheme then achieves the said performance bound under noise-symmetry, that given by the transposition (i,j) where i, j are the free side information. In this correspondence, however, we ob-positions corresponding to the field elements 0, 1. This comes tain a solution for the continuous-time version of the problem from having a codeword with weight two in the dual code. with noisy side information at the receiver and show that our ACKNOWLEDGMENT optimal linear scheme asyptotically achieves the well-known performance bounds in the cases when i) the side information is The author wishes to thank the referees for their careful transmitted to the receiver noise-free, and ii) there is no side reading, comments, and suggestions which have helped greatly information. For the general case, the optimal linear encoder-in improving the presentation of these results. decoder transformations are obtained in closed form. In the next section we formulate the problem and give the REFERENCES solution in Section III. Section IV is devoted to a discussion of 111 E. R. Berlekamp, Algebraic Coding Theory. New York: McGraw-Hill, the asymptotic behavior of the communication system. 1968.