Every binary (2/sup m/-2, 2/sup 2(m)-2-m/, 3) code can be lengthened to form a perfect code of length 2/sup m/-1

Abstract

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 ...

On Optimal Binary One-Error-Correcting Codes of Lengths $2^{m}-4$ and $2^{m}-3$

“…The perfect codes attaining A(15, 3) = 2048 were classified by the second and the third author [14]; the number of equivalence classes of such codes is 5983, with 2165 extensions. Using a result by Blackmore [3], this classification can be used to get the number of equivalence classes of codes attaining A(14, 3) = 1024, which is 38408; these have 5983 extensions. All these results still leave the classification problem open for lengths 12 and 13.…”

“…1 14179 64 8511 2048 39 2 45267 96 90 3072 3 3 41 128 3114 4096 9 4 66449 192 55 6144 4 6 137 256 1247 8192 1 8 44529 384 39 12288 4 12 159 512 403 16384 1 16 32193 768 35 24576 1 24 89 1024 82 73728 1 32 20813 1152 1 147456 1 48 98 1536 15 the unique (13,256,4) code that cannot be lengthened to a (14,512,4) code has an automorphism group of order 384. It turns out that one detail in [5] is incorrect: shortening the (two) (13, 512, 3) codes that cannot be lengthened to (15,2048,3) codes always leads to (12,256,3) codes that cannot be lengthened to (15,2048,3) codes.…”

“…In general, a complete characterization or classification of such codes does not seem feasible, but the classification problem can be addressed for small parameters and general properties of these codes can be studied. For example, the issue whether codes with these parameters can be lengthened to perfect codes has attracted some interest in the literature [3], [4], [5], [6]. For i = 1, every code (2) can be lengthened to a perfect code and this can be done in a unique way up to equivalence [3].…”

“…For example, the issue whether codes with these parameters can be lengthened to perfect codes has attracted some interest in the literature [3], [4], [5], [6]. For i = 1, every code (2) can be lengthened to a perfect code and this can be done in a unique way up to equivalence [3]. Consequently, codes with such parameters are in a direct relationship to the perfect codes, so our main interest is in the codes with i = 2 and i = 3.…”

“…This study is started in Section II by considering certain properties of subcodes, which can be utilized in a computer-aided classification of optimal binary one-error-correcting codes of length 12 and 13, considered in Section III. It turns out that the number of equivalence classes of (12,256,3) and (13,512,3) codes is 237610 and 117823, respectively. Some central properties of the classified codes are analyzed in Section IV.…”