Search

This paper demonstrates two improvements in coding techniques that could be used for memory word coding. First, within the fixed structure of a Hamming single-error-correcting, double-error-detecting (SEC/DED) code, an improvement can be obtained in circuit cost and operational speed over more conventional code implementations. Second, the mechanics of error correction in a fault-tolerant computer may be carried out via conventional hardware means or by use of the existing system facilities, such as the combination of the microprogram unit, local store, and the arithmetic-logic unit. These improvements may be obtained by the use of Rotational Coding schemes in conjunction with a technique calied "lookaside correction." This paper first shows a generalized algorithm for specifying the parity check matrix of Rotational Codes. The structure implemented by the parity check matrix in this paper is not merely encoding and decoding circuitry, but translates between Rotational Code forms and byte-parity encoded forms. The unique feature of these translators is that use of the Rotational Code permits the error correction to be performed on only a subset of the data word bits, and only if a single-error condition has been detected. The correction mechanism may be either a hardware logic circuit or firmware. The paper concludes with a comparison of the circuit requirements and correctional speed of the Hamming (72, 64) single-error-correcting, double-error-detecting code, as it normally would be implemented, and a Rotational Code translator also operating on 64 data bits and 8 check bits.

Let α be a primitive root of G. F. (qm). Let I(m, v) be the number of information symbols of the code with parity check matrix (α\mij), i = 1, …, v, j = 0, …, qm − 2. Let v = qλ, m − λ = r. Then for sufficiently large m we have {fx0153-1} where 〈c〉 denotes the nearest integer to c and ρ is the positive root of the equation xr = (q − 1) (xr−1 + … + 1). For small values of m we have I(m, v) = ρm + ε where | ε \t| ≦ (r − 1)τm, ″>τ 〈< 1. Estimates for τ are given in the Appendix. Two other formulas, exact for all values of m ≧ r − 1, are also given. The first contains r terms, the second [m/r + 1] terms, where [c] denotes the largest integer not exceeding c.

Abramson and Bose and Chakravarti have constructed, simultaneously and independently, binary group codes which correct errors in bursts of three or less for even redundancy. In this paper the corresponding problem when the redundancy is odd is considered. Let r = 2m + 1, m \geqq 4 , and let A be an mxr matrix whose rows are the r -place binary vectors \alpha_1, \alpha_2, \cdots \alpha_n . We show that A satisfies the necessary and sufficient condition for being the parity check matrix of the required code if it is obtained in the following manner: Let x be a primitive element of GF (2^m) , y be a primitive element of GF (2^{m-1}) and z be a primitive element of GF (2^2) . Further suppose 1 + x = x^{\phi}, 1 + y = y^{\phi} where \theta \neq 2 (mod 3) if m is even and \phi \neq 2(mod 3) if m is odd. Let \beta_i be the coefficient vector of the (m-1) -th degree polynomial in x , which represents x^i and let \lambda_i and \rho^i have similar meanings with reference to y^i and z^i . Then \alpha_i = (\beta_i, \lambda_i, \rho_i), i = 0, 1, \cdots (2^m-l) (2^{m--1}-1)- l. Although this method does not yield an optimum n , it is easily shown that as m increases n = (2^m -1) (2^{m-1} -1) approaches the optimal value.

In general, the polynomial whose coefficients are a single right shift of the coefficients of C(z) is z C(z). If C(z) is a polynomial of degree less than n, interpreted as an n-dimensional vector, then the right cyclic shift of C(z) must have Cn−1 as the coefficient of z° and Ci−1 as the coefficient of zi for i = 1, 2,..., n−1. The polynomial which represents the right cyclic shift of C(z) is therefore zC(z) mod, zn−1.

Summary In this paper we investigate some of the properties of augmented Bose-Chaudhuri type parity check matrices. It is shown that some of the ensuing codes not only possess the property of correcting a predetermined number of independent errors, but also corrects certain bursts of errors that may occur in transmission of messages. Using the concept of a cycle it is shown how we may construct Bose-Chaudhuri type parity check matrices for which the number of message places is not of the form (2 m — 1)/ c, where c is an integer greater than or equal to unity.

The paper introduces a super-orthogonal space-time block coding scheme. The code construction is based on the expansion of an orthogonal block code via a unitary matrix transformation. By expanding the orthogonal block code, we can both increase the code rate and improve the performance in terms of E/sub b//N/sub 0/, with a moderate increase of the receiver complexity. The performance is compared with orthogonal and quasi-orthogonal block codes in computer simulation.

It has been shown how arithmetic codes can be embedded in the structure of a module and how generator and check matrices may be defined. The motivation is to parallel the analogy with communication codes with the purpose of showing majority decoding. The technical difficulties in the derivations are pointed out while some partial results are given.