Codes C 1 , . . . , C M of length n over F q and an M × N matrix A over F q define a matrix-product codeWe study matrix-product codes using Linear Algebra. This provides a basis for a unified analysis of |C|, d(C), the minimum Hamming distance of C, and C ⊥ . It also reveals an interesting connection with MDS codes. We determine |C| when A is non-singular. To underbound d(C), we need A to be 'non-singular by columns (NSC)'. We investigate NSC matrices. We show that Generalized Reed-Muller codes are iterative NSC matrix-product codes, generalizing the construction of Reed-Muller codes, as are the ternary 'Main Sequence codes'. We obtain a simpler proof of the minimum Hamming distance of such families of codes. If A is square and NSC, C ⊥ can be described using C ⊥ 1 , . . . , C ⊥ M and a transformation of A. This yields d(C ⊥ ). Finally we show that an NSC matrix-product code is a generalized concatenated code.

We calculate the cross sections for the production of the J/+ in nucleon-nucleon interactions on the basis of a naive Drell-Yan-type parton model where the I production is considered as production through the resonant or fusion interaction of quarks and antiquarks resident in the colliding nucleons. The interacting quarks are assumed to act as free point Dirac particles obeying the Bethe-Heitler description of annihilation to lepton pairs. Unitarity limits on I production established by calcutations conducted on this basis are shown tt be much smaller than the measured cross section, indicating that such models are unsatisfactory. Similar considerations suggest that the 1 production cannot be described by the interactions of nearly free gluons. where Fq is the width for a specific quark, FM, is the width for decay to muon pairs, and F is the total width. We then find the number of resonant interactions per second derived from the flux A(M) by integrating Eq. 2 with the I-spin, J = 1, and the quark spin, Sq = 1/2: Nil = A(M)7,M(M) dM = (6Ir2rqFJ A(M,))/M*2. [3] The ratio of resonance production of muon pairs and the background production through the Bethe-Heitler mechanism will then be: RJ= =A~b 9IrrFMQ4 /Nbh =2a2rAMQ2 [4] For a given value of FM, this ratio will be largest for Fq F» >> rF; for a given value of r, the ratio will be greatest for the condition r, = Fq = F/2, which corresponds to the familiar unitary limit of ir/k2 with spin factors. The respective limits will then be: RJM = 2.66-1o5Fr/(AM Q2) or RAM = 0.665-i0 rF/(AM Q2), [5] where Q2 cannot be less than 1/9. We can define a more realistic limit by considering that the mean square charge effective in background pair production in proton-nucleus interactions will be about 1/3, and from 1A-e universality: r, = re. If we introduce color and assume that u and d-quark couplings to the resonant state are equal, the maximum possible value of the ratio RM will occur for the condition such that rur=ruw=rub=Frd=rdw=rdb=r/6>>r =re and, for numerical convenience, taking AM as 1 GeV/c2 and r in keV, RM = 0.13 FM. [6] If we express the maximum value of R, in terms of the total width, the maximum will occur under the condition FlFre=3rur=... =r/4 and R.] = 0.021 F. [7] If we use the measured* value of FM for the I as 4.8 keV, the maximum value of RJMallowed by the unitary limit of Eq. 6 will be 0.64 independent of the total width. If we use the measured total width, F = 69 keV, as a constraint, the maximum value of Ri', calculated from Eq. 7, will be 1.45 independent of the magnitude of r,. We will see that the measured value of R, is of the order of, or greater than, 5 and then violates these limits by large factors. Most of the reports of measurements of J/I production in nucleon-nucleon interactions through measurements of lepton pairs do not present values of the continuum in a manner that can be applied to an evaluation of RI, unambiguously. We then present measured valuest of Ba in Fig. 1 together with the "best fit" to the continuum lepton pair production ...

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We study trellises of Reed}Muller codes from "rst principles. Our approach to local trellis behaviour seems to be new and yields amongst other things another proof of a result of Berger and Be'ery on the state complexity of Reed}Muller codes. We give a general form of a minimal-span generator matrix for the family of Reed}Muller codes with their standard bit-order. We apply this to determining the number of parallel subtrellises in any uniform sectionalisation of a Reed}Muller code and to designing trellises for Reed}Muller codes with more parallel subtrellises than the minimal trellis, but with the same state complexity. Academic Press

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