coefficient that is adjacent, in a given polynomial, to one whose value is nonzero, and the set of its partial derivatives with respect to some or all possible virtual elements is calculated; a nonzero partial derivative indicates that the zero-valued coefficient will become nonzero if the corresponding virtual element is replaced by a real one of the same type. In fact, by considering the partial derivatives of more than one zero- valued coefficient, it is sometimes possible to simultaneously correct a number of these coefficients by the introduction of only one element (first and second changes of Fig. 1). It is sound policy to prohibit, as far as possible, connections between external nodes at this stage, as the encouragement of the growth of connections between internal nodes has been observed to generally speed up the evolutionary process. (b) Optimisation carried out at each stage reached in the evolutionary process, i.e. with fixed network topology, takes place in the domain of the square of the independent variables. This constrains the element values to be positive during the optimisation procedure, and also indicates, by their being driven to very low values, which elements might be removed from the network. The removal of elements occurs at the fifth change in the example shown. In every case, the optimisation process is continued until significant changes in element values etc. cease to occur. (c) The following algorithm was used to determine the type of network element to grow and also its position in the network and its initial value: with the elements already present in the network fixed in value and all possible virtual elements regarded as independent variables, one stage of Levenberg's optimisation algorithm was performed in the domain of these variables for a number of positive values of the Levenberg parameter X. A very wide range of the parameter X was used, with equal intervals on a logarithmic X scale, the vector of corrections to the variables was calculated for each of these values of X and the component of this vector having the maximum positive value was noted. This particular com- ponent, corresponding to a positive virtual element, was then incorporated with the fixed-value elements already present in the network, and an objective function, equal to the sum of the squares of the errors used throughout, was calculated. This objective function, as a function of X, is usually discontinuous and multimodal, since, over the wide range of X used, different components of the Levenberg correction vector possess the largest positive value. A golden-section linear search, of relatively low accuracy, was then performed between adjacent values of the parameter X used, followed by a final, more accurate, search over the region of X associated with the best minimum thus obtained. Growing elements by this method takes place at the third and fourth stages in the example shown. The example shown in Fig. 1 is only one of several twin-T RC structures, each of which has been successfully synthetised from different types of RC ladder network. Taken together, these examples demonstrate that an evolutionary approach to linear-network synthesis is now feasible, at any rate, for problems of this order of difficulty. Acknowledgment: P. H. di Mambro would like to express his gratitude to Prof. J. F. Coales for the use of computing facilities within the Department of Engineering, University of Cambridge, England. o. P. D. CUTTERIDGE 27th December 1973 Department of Engineering University of Leicester Leicester LEI 7RH, England P. H. DI MAMBRO Department of Engineering & Building Cambridgeshire College of Arts & Technology, Collier Road, Cambridge CB1 2AJ, England References 1 CALAHAN, D. A.: 'Computer design of linear frequency selective networks', Proc. Inst. Elect. Electron. Eng., 1965, 53, pp. 1701-1706 2 DIRECTOR, s. w., and ROHRER, R. A.: 'Automated network design—the frequency domain case', IEEE Trans., 1969, CT-16, pp. 330-337 3 DIRECTOR, s. w.: 'Survey of circuit-oriented optimization techniques', ibid., 1971, CT-18, pp. 3-10 4 BOWN, c c. s., and GEIGER, G. V.: 'Design and optimisation of circuits by computer', Proc. IEE, 1971, 118, (5), pp. 649-661 5 CUTTERIDGE, o. p. D.: 'Computer synthesis of lumped linear networks of arbitrary structure' in SKWIRZYNSKI, J. K., and SCANLAN, J. O. (Eds.): 'Network and signal theory' (Peter Peregrinus, 1973), pp. 105-111 6 CUTTERIDGE, o. p. D., and DI MAMBRO, P. H.: 'Simultaneous generation of the coefficients of network polynomials and their partial derivatives from the nodal-admittance matrix', Electron. Lett., 1970, 6, pp. 308- 310 7 CUTTERIDGE, o. p. D., and DI MAMBRO, p. H.: 'Simultaneous generation of the partial derivatives of network polynomial coefficients: further details and results, ibid., 1971, 7, pp. 3-4 8 LEVENBERG, K.: 'A method for the solution of certain non-linear problems in least squares', Quart. J. Appl. Math., 1944, 2, pp. 164-168 SOME NEW RESULTS ON BINARY LINEAR BLOCK CODES Indexing term: Error-correction codes Certain properties of the parity-check matrix H of (n, k) linear codes are used to establish a computerised search procedure for new binary linear codes. Of the new error-correcting codes found by this procedure, two codes were capable of correcting up to two errors, three codes up to three errors, four codes up to four errors and one code up to five errors. Two meet the lower bound given by Helgert and Stinaff, and seven codes exceed it. In addition, one meets the upper bound. Of the even-Hamming-distance versions of these codes, eight meet the upper bound, and the remaining two exceed the lower bound. Introduction: A linear block (n, k, i) error-correcting code 2 - 3 comprises k distinct codewords, q being the number of symbols per sign; for binary codes, q = 2, which form a subspace V of the vector space V n over thefieldF of q elements. The basis vectors of the subspace V can be considered to be the rows of a matrix G, called the generator matrix of V. The basis vectors of the null space V' of the subspace V can be considered as the rows of the parity-check matrix //. Since V' is the row space of H and the null space of V, a vector v is in V if and only if it is orthogonal to every row of H. It follows that, for each codeword of Hamming weight co, there is a linear-dependence relation between co columns of//; conversely, for each linear-dependence relation involving co columns of H, there is a codeword of weight co. 2 In general, we can say that, if an (n, k) code V has a parity-check matrix H, V will correct all errors of weight t or less, if and only if every It columns from H are linearly independent. 2 The ELECTRONICS LETTERS 7th February 1974 Vol.10 No. 3 parity-check matrix may be viewed in a slightly different, but nevertheless useful way, as follows: Let a subset U of n vectors in a vector space F n _ k be formed over a field F of q elements, and let U contain at least one set of the basis vectors of F n _ k . A parity-check matrix H can be produced so that its n columns are the vectors in U. The (n, k) linear code corresponding to this matrix will correct t random errors if and only if all linear combinations of every t vectors in U give unique nonzero vectors in V n _ k . That this statement is correct can be seen from the following argument: Since the set U includes at least one set of the basis vectors of the vector space F n _ k , the parity-check matrix //, whose columns are the vectors of U, has a rank equal to its dimen- sion. All the n — k rows of H are thus linearly independent. Now consider It vectors of the set U. Let these 2f vectors form two sets U t and U 2 of t vectors each. The linear com- bination of all vectors of l/i form a set S x of q* unique vectors. Similarly, set S 2 corresponds to the linear com- binations of all vectors of U 2 . Since Si contains all the vectors that are the linear combinations of the vectors (u u u 2 , ...,u t )eU u it follows that, if a vector J is in Si, all vectors of the scalar product (as) are in Si also, where a= 1,2, ...,q— 1. This implies that the modulo-g addition of any vector in S x with any vector in S 2 is a nonzero vector, and therefore every It vectors of U are linearly independent. As a consequence, every It columns of the parity-check matrix H are linearly independent, and hence the corres- ponding (n, k) code can correct up to t random errors. 31