45 research outputs found
The Pace code, the Mathieu group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml1" display="inline" overflow="scroll" altimg="si1.gif"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub></mml:math> and the small Witt design <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml2" display="inline" overflow="scroll" altimg="si2.gif"><mml:mi>S</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mn>6</mml:mn><mml:mo>,</mml:mo><mml:mn>12</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math>
No full textA ternary [66,10,36]3-code admitting the Mathieu group M12 as a group of automorphisms has recently been constructed by N. Pace, see Pace (2014). We give a construction of the Pace code in terms of M12 as well as a combinatorial description in terms of the small Witt design, the Steiner system S(5,6,12). We also present a proof that the Pace code does indeed have minimum distance 36
Orbits of Lines for a Twisted Cubic in PG (3 , q)
No full textIn the projective space PG (3 , q) , we consider the orbits of lines under the stabilizer group of the twisted cubic. In the literature, lines of PG (3 , q) are partitioned into classes, each of which is a union of line orbits. In this paper, all classes of lines consisting of a unique orbit are found. For the remaining line types, with one exception, it is proved that they consist exactly of two or three orbits; sizes and structures of these orbits are determined. Also, the subgroups of the stabilizer group of the twisted cubic fixing lines of the orbits are obtained. Problems which remain open for one type of lines are formulated and, for 5 ≤ q≤ 37 and q= 64 , a solution is provided
Optimal Additive Quaternary Codes of Low Dimension
No full textAn additive quaternary [n, k, d]-code (length n, quaternary dimension k, minimum distance d) is a 2kdimensional F2-vector space of n-Tuples with entries in F2-F2 (the 2-dimensional vector space over F2) with minimum Hamming distance d. We determine the optimal parameters of additive quaternary codes of dimension k-3. The most challenging case is dimension k = 2.5. We prove that an additive quaternary [n, 2.5, d]-code whered n-1 exists if and only if 3(n-d) d/2+d/4+d/8. In particular, we construct new optimal 2.5-dimensional additive quaternary codes. As a by-product, we give a direct proof for the fact that a binary linear [3m, 5, 2e]2-code for e m-1 exists if and only if the Griesmer bound 3(m-e) e/2+e/4+e/8 is satisfied.
ON THE WEIGHT DISTRIBUTION OF THE COSETS OF MDS CODES
No full textThe weight distribution of the cosets of maximum distance separable (MDS) codes is considered. In 1990, P.G. Bonneau proposed a relation to obtain the full weight distribution of a coset of an MDS code with minimum distance d using the known numbers of vectors of weights ≤ d − 2 in this coset. In this paper, the Bonneau formula is transformed into a more structured and convenient form. The new version of the formula allows to consider effectively cosets of distinct weights W . (The weight W of a coset is the smallest Hamming weight of any vector in the coset.) For each of the considered W or regions of W, special relations more simple than the general ones are obtained. For the MDS code cosets of weight W = 1 and weight W = d − 1 we obtain formulas of the weight distributions depending only on the code parameters. This proves that all the cosets of weight W = 1 (as well as W = d − 1) have the same weight distribution. The cosets of weight W = 2 or W = d − 2 may have different weight distributions; in this case, we proved that the distributions are symmetrical in some sense. The weight distributions of the cosets of MDS codes corresponding to arcs in the projective plane PG(2, q) are also considered. For MDS codes of covering radius R = d − 1 we obtain the number of the weight W = d − 1 cosets and their weight distribution that gives rise to a certain classification of the so-called deep holes. We show that any MDS code of covering radius R = d − 1 is an almost perfect multiple covering of the farthest-off points (deep holes); moreover, it corresponds to an optimal multiple saturating set in the projective space PG(N, q)
On Cosets Weight Distribution of Doubly-Extended Reed-Solomon Codes of Codimension 4
No full textWe consider the 3 generalized doubly-extended Reed-Solomon code of codimension 4 as the code associated with the twisted cubic in the projective space mathrm {PG}(3, ext {q}). Basing on the point-plane incidence matrix of mathrm {PG}(3, ext {q}) , we obtain the number of weight 3 vectors in all the cosets of the considered code. This allows us to classify the cosets by their weight distributions and to obtain these distributions. The weight of a coset is the smallest Hamming weight of any vector in the coset. For the cosets of equal weight having distinct weight distributions, we prove that the difference between the w-th components, 3 < ext {w}le ext {q}+1 , of the distributions is uniquely determined by the difference between the 3-rd components. This implies an interesting (and in some sense unexpected) symmetry of the obtained distributions
Additive Quaternary Codes Related to Exceptional Linear Quaternary Codes
No full textWe study additive quaternary codes whose parameters are close to those of the extended cyclic [12,6,6]_4-code or to the quaternary linear codes generated by the elliptic quadric in PG(3,4) or its dual. In particular we characterize those codes in the category of additive codes and construct some additive codes whose parameters are better than those of any linear quaternary code. Our new code parameters are [22,17.5,4]_4
Orbits of the Class O6 of Lines External to the Twisted Cubic in PG (3 , q)
No full textIn the projective space PG (3 , q) , we consider orbits of lines under the stabilizer group of the twisted cubic. In the literature, lines of PG (3 , q) are partitioned into classes, each of which is a union of line orbits. We propose an approach to obtain orbits of the class named O6, whose complete classification is an open problem. For all q, we describe a family of orbits of O6 and their stabilizer groups. The orbits of this family include an essential part of all O6 orbits
Twisted cubic and point-line incidence matrix in PG (3 , q)
No full textWe consider the structure of the point-line incidence matrix of the projective space PG (3 , q) connected with orbits of points and lines under the stabilizer group of the twisted cubic. Structures of submatrices with incidences between a union of line orbits and an orbit of points are investigated. For the unions consisting of two or three line orbits, the original submatrices are split into new ones, in which the incidences are also considered. For each submatrix (apart from the ones corresponding to a special type of lines), the numbers of lines through every point and of points lying on every line are obtained. This corresponds to the numbers of ones in columns and rows of the submatrices
On blocking sets of inversive planes
No full textLet S be a blocking set in an inversive plane of order q. It was shown by Bruen
and Rothschild [1] that |S| >= 2q for q >= 9. We prove that if q is sufficiently large, C is a fixed
natural number and |S| = 2q + C, then roughly 2/3 of the circles of the plane meet S in one
point and 1/3 of the circles of the plane meet S in four points. The complete classification
of minimal blocking sets in inversive planes of order q <= 5 and the sizes of some examples
of minimal blocking sets in planes of order q <= 37 are given. Geometric properties of
some of these blocking sets are also studied
