Properties of Error-Correcting Codes at Low Signal-To-Noise Ratios

• Published in: SIAM journal on applied mathematics 1967-07, Vol.15 (4), p.775-798, Article 775
• Main Author: Posner, Edward C.
• Format: Article
• Language: English
• Subjects: Binary codes; Codes; Coding theory; Cyclic codes; Decryption; Error correcting codes; Error correction & detection; Integers; Polynomials; Power gain; Power loss; Probability; Ratios; Signal to noise ratio
• ISSN: 0036-1399; 1095-712X
• DOI: 10.1137/0115068

In this paper, the use of error-correcting codes on a white Gaussian channel is considered as the signal-to-noise ratio approaches zero. The system with which a fixed code is compared is the system which does not code but preserves the information rate. Two criteria of performance are used: expected number of information bits in error, and probability of word error. It is shown that if bit-by-bit detection is used with a fixed code, and if the expected number of bits in error is to be minimized, then any decoding scheme results in a loss at low signal-to-noise ratios. In fact, the best decoding scheme in this context often becomes the one which ignores the check symbols entirely. On the other hand, if word error probability is to be minimized, then error-correcting codes used with bit-by-bit detection can yield only slight power gain and usually result in a loss. However, orthogonal codes using correlation detection give a gain, as is well known, and this gain approaches 3.4 db as the length of the code increases without bound. But even orthogonal codes used with correlation detection result in a loss at low signal-to-noise ratios if the expected number of information bits in error is the criterion.