On linear codes whose weights and length have a common divisor

Designs, Codes and Cryptography, 2008

In this paper, we study the p-ary linear code C(PG(n,q)), q = p h , p prime, h ≥ 1, generated by the incidence matrix of points and hyperplanes of a Desarguesian projective space PG(n,q), and its dual code. We link the codewords of small weight of this code to blocking sets with respect to lines in PG(n,q) and we exclude all possible codewords arising from small linear blocking sets. We also look at the dual code of C(PG(n,q)) and we prove that finding the minimum weight of the dual code can be reduced to finding the minimum weight of the dual code of points and lines in PG(2,q). We present an improved upper bound on this minimum weight and we show that we can drop the divisibility condition on the weight of the codewords in Sachar’s lower bound (Geom Dedicata 8:407–415, 1979).

Designs, Codes and Cryptography, 2007

We determine the minimum length n q (k, d) for some linear codes with k ≥ 5 and q ≥ 3. We prove that n q (k, d) = g q (k, d) + 1 for q k−1 − 2q k−1 2 −q + 1 ≤ d ≤ q k−1 − 2q k−1 2 when k is odd, for q k−1 − q k 2 − q k 2 −1 − q + 1 ≤ d ≤ q k−1 − q k 2 − q k 2 −1 when k is even, and for 2q k−1 − 2q k−2 − q 2 − q + 1 ≤ d ≤ 2q k−1 − 2q k−2 − q 2 .

We obtain, in principle, a complete classification of all long inextendable binary linear codes. Several related constructions and results are presented. LEMMA 2.3 If X is periodic then |X | is even.

Designs, Codes and Cryptography, 2019

An outstanding folklore conjecture asserts that, for any prime p, up to isomorphism the projective plane P G(2, F p) over the field F p := Z/pZ is the unique projective plane of order p. Let π be any projective plane of order p. For any partial linear space X , define the inclusion number i(X , π) to be the number of isomorphic copies of X in π. In this paper we prove that if X has at most log 2 p lines, then i(X , π) can be written as an explicit rational linear combination (depending only on X and p) of the coefficients of the complete weight enumerator (c.w.e.) of the p-ary code of π. Thus, the c.w.e. of this code carries an enormous amount of structural information about π. In consequence, it is shown that if p > 2 9 = 512, and π has the same c.w.e. as P G(2, F p), then π must be isomorphic to P G(2, F p). Thus, the uniqueness conjecture can be approached via a thorough study of the possible c.w.e. of the codes of putative projective planes of prime order.

Discrete Mathematics, 1985

s), s = 2 ~ and n >12, denote the Desarguesian projective space of projective dimension n over the Galois field Fs. The set of its subsets with set theoretic symmetric difference as addition is a vector space over F2. For 1 ~< t n-1, let Ct(n, s) denote its subspace generated by the t-flats of PG(n, s) and for w c_ PG(n, s), let [wl denote the cardinality (or weight) of w. Our object in this note is to present a purely geometric proof of the following theorem proved independently by Smith [5] and Delsarte et al. [2]. Theorem. For s = 2", n > 1 and 0<t<n, the words of Ct(n, s) of least non-zero weight are precisely the t-fiats of PG(n, s). Some crucial parts of the proof are contained in the following lemmas.

Designs, Codes and Cryptography, 2011

A set of n + k points (k > 0) in projective space of dimension n is said to be an (n + k)-arc if there is no hyperplane containing any n + 1 points of the set. It is well-known that for the projective space PG(n, q), this is equivalent to a maximum distance separable linear code with symbols in the finite field GF(q), of length n + k, dimension n + 1, and distance d = k that satisfies the Singleton bound d ≤ k. We give an algebraic condition for such a code, or set of points, and this is associated with an identity involving determinants.

Finite Fields and Their Applications, 2018

Consider the Grassmann graph formed by k-dimensional subspaces of an n-dimensional vector space over the field of q elements (1 < k < n − 1) and denote by Π(n, k)q the restriction of this graph to the set of projective [n, k]q codes. In the case when q ≥ n 2 , we show that the graph Π(n, k)q is connected, its diameter is equal to the diameter of the Grassmann graph and the distance between any two vertices coincides with the distance between these vertices in the Grassmann graph. Also, we give some observations concerning the graphs of simplex codes. For example, binary simplex codes of dimension 3 are precisely maximal singular subspaces of a non-degenerate quadratic form.

IEEE Transactions on Information Theory, 2000

Some new infinite families of short quasi-perfect linear codes are described. Such codes provide improvements on the currently known upper bounds on the minimal length of a quasi-perfect [n; n 0m; 4] -code when either 1) q = 16; m 5; m odd, or 2) q = 2 ; 7 i 15; m 4, or 3) q = 2 ;` 8; m 5; m odd. As quasi-perfect [n; n0m; 4] -codes and complete n-caps in projective spaces P G(m 01;q) are equivalent objects, new upper bounds on the size of the smallest complete cap in P G(m 01;q) are obtained.

Journal of Geometry, 1997

We develop a technique for improving the universal linear programming bounds on the cardinality and the minimum distance of codes in projective spaces I FP n−1 . We firstly investigate test functions P j (m, n, s) having the property that P j (m, n, s) < 0 for some j if and only if the corresponding universal linear programming bound can be further improved by linear programming. Then we describe a method for improving the universal bounds. We also investigate the possibilities for attaining the first universal bounds.

IEEE Transactions on Information Theory, 2000

The projective space of order n over the finite field q , denoted here as Pq(n), is the set of all subspaces of the vector space n q . The projective space can be endowed with the distance function d(U; V ) = dim U + dim V 0 2 dim(U \V ) which turns Pq(n) into a metric space. With this, an (n; M; d) code in projective space is a subset of Pq(n) of size M such that the distance between any two codewords (subspaces) is at least d. Koetter and Kschischang recently showed that codes in projective space are precisely what is needed for error-correction in networks: an (n; M; d) code can correct t packet errors and packet erasures introduced (adversarially) anywhere in the network as long as 2t + 2 < d. This motivates our interest in such codes. In this paper, we investigate certain basic aspects of "coding theory in projective space." First, we present several new bounds on the size of codes in P q (n), which may be thought of as counterparts of the classical bounds in coding theory due to Johnson, Delsarte, and Gilbert-Varshamov. Some of these are stronger than all the previously known bounds, at least for certain code parameters. We also present several specific constructions of codes and code families in P q (n). Finally, we prove that nontrivial perfect codes in P q (n) do not exist.

Ieee Transactions on Information Theory, 2004

Infinite families of linear codes with covering radius = 2, 3 and codimension + 1 are constructed on the base of starting codes with codimension 3 and 4. Parity-check matrices of the starting codes are treated as saturating sets in projective geometry that are obtained by computer search using projective properties of objects. Upper bounds on the length function and on the smallest sizes of saturating sets are given.

IEEE Transactions on Information Theory, 2000

IEEE Transactions on Information Theory, 1999

New constructions of linear nonbinary codes with covering radius R = 2 are proposed. They are in part modifications of earlier constructions by the author and in part are new. Using a starting code with R = 2 as a "seed" these constructions yield an infinite family of codes with the same covering radius. New infinite families of codes with R = 2 are obtained for all alphabets of size q 4 and all codimensions r 3 with the help of the constructions described. The parameters obtained are better than those of known codes. New estimates for some partition parameters in earlier known constructions are used to design new code families. Complete caps and other saturated sets of points in projective geometry are applied as starting codes. A table of new upper bounds on the length function for q = 4; 5; 7; R = 2; and r 24 is included.

be the smallest integer n for which there exists a linear code of length n, dimension IC, and minimum distance d, over a field of q elements. In this correspondence we determine n5 (4, d ) for all but 22 values of d. Index Terms-Optimal q-ary linear codes, minimum-length bounds. Publisher Item Identifier S 0018-9448(97)00108-9.

2007

In this paper, we study the p-ary linear code C(PG(n,q)), q = ph, p prime, h 1, generated by the incidence matrix of points and hyperplanes of a Desarguesian projective space PG(n,q), and its dual code. We link the codewords of small weight of this code to blocking sets with respect to lines in PG(n,q) and we exclude all possible