IEEE COMMUNICATIONS LETTERS, VOL. 3, NO. 11, NOVEMBER 1999 317 Convolutional Codes with Optimum Distance Spectrum P˚ al Frenger, P˚ al Orten, and Tony Ottosson, Member, IEEE Abstract—New convolutional codes with rates , , and are presented for constraint lengths ranging from to . These new codes are maximum free-distance codes. Furthermore, the codes have optimized information error weights, resulting in a low bit-error rate for binary communication on both additive white Gaussian noise (AWGN) and Rayleigh fading channels. Improved coding gains of as much as 0.6 dB compared to previously published codes have been observed for coherent BPSK over a Rayleigh fading channel and a wide range of signal-to-noise ratios. Index Terms— Convolutional codes, optimum distance spec- trum. I. INTRODUCTION G OOD convolutional codes are normally found by ex- tensive computer searches. However, the number of possible codes increases exponentially with both the constraint length and the number of generator polynomials. The first attempts to find good codes terminated the search as soon as a code fulfilling the Heller bound was found [1]. Since, the Heller bound is an upper bound on the free distance, the resulting codes were denoted maximum free distance (MFD) codes. The free distance of a convolutional code provides a first- order asymptotic approximation of the error performance. A better estimate of the bit-error rate is obtained by the union upper bound on the bit-error probability, given by [2, p. 488] (1) where is the sum of bit errors (the information error weight) for error events of distance and is the free distance of the code. is the pairwise error probability, given by [2, p. 487] (2) for an AWGN channel. Here, represents the code rate, denotes the received energy per information bit, is the double-sided power spectral density of the noise process, and Observe that a low value Manuscript received July 13, 1999. The associate editor coordinating the review of this letter and approving it for publication was Prof. N. C. Beaulieu. P. Frenger is with Ericsson Research, Ericsson Radio Systems AB, SE-164 80 Stockholm, Sweden. P. Orten and T. Ottosson are with the Communication Systems Group, Department of Signals and Systems, Chalmers University of Technology, SE- 412 96 G¨ oteborg, Sweden. Publisher Item Identifier S 1089-7798(99)08842-0. Fig. 1. Upper bound on the bit-error rate on an uncorrelated flat Rayleigh fading channel for optimum distance spectrum (ODS) and maximum free-distance (MFD) codes. results in low bit-error rate, but a small increase in is enough to compensate for a high value. On a Rayleigh fading channel, the pairwise error probability is given by [2, p. 781] (3) with (4) where is the average It can be seen that for the fading channel decreases slower than for the AWGN case. Thus the information error weight will have larger influence on the bit-error rate. It is reasonable to conclude that codes that perform well on both AWGN and Rayleigh fading channels should have maximum free distance but also a low information error weight on each error path. In what follows, we formalize these conditions into the optimum distance spectrum criterion. We then use this criterion to find codes that are suited for binary transmission over both AWGN and Rayleigh fading channels. II. OPTIMUM DISTANCE SPECTRUM CODES We define an optimum distance spectrum (ODS) convolu- tional code as a code generated by a feedforward encoder 1089–7798/99$10.00  1999 IEEE