 of a positive or negative acknowledgement (ACK or NAK) on receipt of a data block; this is known as stop-and-wait ARQ (Automatic Repeat Request). In go-back-N ARQ, receipt of a NAK by the transmitter makes it retransmit the erroneous codeword and the N - 1 that follow, where N is chosen SO that the time taken to send N codewords is less than the round-trip delay from transmitter to receiver and back again. This obviates the need for a buffer at the receiver. In selective-repeat ARQ, only the codewords for which NAKs have been returned are retransmitted. Performance analysis (Lin & Costello, 1983) Principles of ECC shows that this is the most efficient system, although it requires an adequately large buffer in the receiver. Arithmetic for ECC In a digital communication system, each transmitted digit is selected from a finite set of M values and is described as an M-ary digit. For example, binary digits (bits) have one of two values, which can be represented as 0 and 1. (The actual values of the physical signal used to transmit the digits-for example, voltages-are irrelevant here.) If the message digits in the transmitter do not use the same value of M as the transmitted digits, they are first con- verted to M-ary digits before coding. The analytical design of coders and decoders for M-ary digits requires the use of Galois field arithmetic, denoted GF(M). If M is a prime number (including binary digits, where M = 2), Galois field arithmetic is equivalent to arithmetic modulo-M. GF(M) arithmetic also exists when M is equal to a power of a prime-for example, for quaternary (4-valued) and octal (8-valued) digit systems-but then GF(M) arithmetic is not equivalent to modulo-M arithmetic. Modulo-2 addition is equivalent to the logical exclusive OR (XOR) function, and mul- tiplication is equivalent to the logical AND function. In modulo-2 arithmetic, addition and subtraction are equivalent. Some textbooks only discuss binary coding and therefore treat all subtractions as additions. However, for values of M other than 2 a