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Register a new userMaximum Distance Separable Codes and Arcs in Projective Spaces
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Authors
T.L. Alderson
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Given any linear code $C$ over a finite field $GF(q)$ we show how $C$ can be described in a transparent and geometrical way by using the associated Bruen-Silverman code. Then, specializing to the case of MDS codes we use our new approach to offer improvements to the main results currently available concerning MDS extensions of linear MDS codes. We also sharply limit the possibilities for constructing long non-linear MDS codes.
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For which positive integers $n,k,r$ does there exist a linear $[n,k]$ code $C$ over $mathbb{F}_q$ with all codeword weights divisible by $q^r$ and such that the columns of a generating matrix of $C$ are projectively distinct? The motivation for studying this problem comes from the theory of partial spreads, or subspace codes with the highest possible minimum distance, since the set of holes of a partial spread of $r$-flats in $operatorname{PG}(v-1,mathbb{F}_q)$ corresponds to a $q^r$-divisible code with $kleq v$. In this paper we provide an introduction to this problem and report on new results for $q=2$.
A projective linear code over $mathbb{F}_q$ is called $Delta$-divisible if all weights of its codewords are divisible by $Delta$. Especially, $q^r$-divisible projective linear codes, where $r$ is some integer, arise in many applications of collections of subspaces in $mathbb{F}_q^v$. One example are upper bounds on the cardinality of partial spreads. Here we survey the known results on the possible lengths of projective $q^r$-divisible linear codes.
We formulate an Algebraic-Coding Equivalence to the Maximum Distance Separable Conjecture. Specifically, we present novel proofs of the following equivalent statements. Let $(q,k)$ be a fixed pair of integers satisfying $q$ is a prime power and $2leq k leq q$. We denote by $mathcal{P}_q$ the vector space of functions from a finite field $mathbb{F}_q$ to itself, which can be represented as the space $mathcal{P}_q := mathbb{F}_q[x]/(x^q-x)$ of polynomial functions. We denote by $mathcal{O}_n subset mathcal{P}_q$ the set of polynomials that are either the zero polynomial, or have at most $n$ distinct roots in $mathbb{F}_q$. Given two subspaces $Y,Z$ of $mathcal{P}_q$, we denote by $langle Y,Z rangle$ their span. We prove that the following are equivalent. [A] Suppose that either: 1. $q$ is odd 2. $q$ is even and $k otin {3, q-1}$. Then there do not exist distinct subspaces $Y$ and $Z$ of $mathcal{P}_q$ such that: 3. $dim(langle Y, Z rangle) = k$ 4. $dim(Y) = dim(Z) = k-1$. 5. $langle Y, Z rangle subset mathcal{O}_{k-1}$ 6. $Y, Z subset mathcal{O}_{k-2}$ 7. $Ycap Z subset mathcal{O}_{k-3}$. [B] Suppose $q$ is odd, or, if $q$ is even, $k otin {3, q-1}$. There is no integer $s$ with $q geq s > k$ such that the Reed-Solomon code $mathcal{R}$ over $mathbb{F}_q$ of dimension $s$ can have $s-k+2$ columns $mathcal{B} = {b_1,ldots,b_{s-k+2}}$ added to it, such that: 8. Any $s times s$ submatrix of $mathcal{R} cup mathcal{B}$ containing the first $s-k$ columns of $mathcal{B}$ is independent. 9. $mathcal{B} cup {[0,0,ldots,0,1]}$ is independent. [C] The MDS conjecture is true for the given $(q,k)$.
A projective Reed-Muller (PRM) code, obtained by modifying a (classical) Reed-Muller code with respect to a projective space, is a doubly extended Reed-Solomon code when the dimension of the related projective space is equal to 1. The minimum distance and dual code of a PRM code are known, and some decoding examples have been represented for low-dimensional projective space. In this study, we construct a decoding algorithm for all PRM codes by dividing a projective space into a union of affine spaces. In addition, we determine the computational complexity and the number of errors correctable of our algorithm. Finally, we compare the codeword error rate of our algorithm with that of minimum distance decoding.
In this paper new infinite families of linear binary completely transitive codes are presented. They have covering radius $rho = 3$ and 4, and are a half part of the binary Hamming and the binary extended Hamming code of length $n=2^m-1$ and $2^m$, respectively, where $m$ is even. From these new completely transitive codes, in the usual way, i.e., as coset graphs, new presentations of infinite families of distance transitive coset graphs of diameter three and four, respectively, are constructed.
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