Convolutional Reed-Solomon Codes

This paper is concerned with a family of character error correcting convolutional codes which are derived from Reed-Solomon block codes. 1 T h e derivation of the error correcting capabilities is easy because of the algebraic n a t u r e of the code; t h e convolutional nature of the code allows the use of a simple encoder even at high rates. Also, sequential decoding techniques might be applicable. Throughout the paper we use elements of a finite field as symbols 729 730 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1969 instead of just 1 and 0. T h e elements can be represented by fc-tuples of ones and zeros if the field has 2 k elements; these k tuples are called symbols or characters. T h u s with this code we are able to correct character errors which m a y be bursts of binary errors. All t h a t one need know about finite fields is t h a t each nonzero element has an inverse and t h a t there exists at least one element which when t a k e n to successive powers will generate the entire field with the exception of the zero element. This is called a primitive element. T h e capability of the codes is given by the number of errors t h a t can be corrected within a fixed number of characters (the constraint length). Suppose a code can correct three errors within a constraint length of 12. T h e n the code can correct any p a t t e r n of errors as long as no sequence of 12 characters has more t h a n three errors in it. In the context of error correcting ability one can define a minimum distance of the code.