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A combinatorial problem for vector spaces over finite fields. (English) Zbl 0747.11063

Let \(V\) be a vector space over an arbitrary field of finite dimension \(m\geq 1\) and let \(s\) and \(m_ 1,m_ 2,\ldots,m_ s\) be positive integers. For a system \(C\) of vectors of the form \(c_ j^{(i)}\in V\), for \(1\leq j\leq m_ i\), \(1\leq i\leq s\) define \(\rho(C)=\min\sum^ s_{i=1}d_ i\) where the minimum is extended over all integers \(d_ 1,\ldots,d_ s\) with \(0\leq d_ i\leq m_ i\) for \(1\leq i\leq s\) and \(\sum^ s_{i=1}d_ i\geq 1\) for which the subsystem \(\{c_ j^{(i)}:1\leq j\leq d_ i\), \(1\leq i\leq s\}\) is linearly dependent in \(V\). If there are no such \(d_ 1,\ldots,d_ s\) then define \(\rho(C)=m+1\). This paper studies the number \(R(V;m_ 1,\ldots,m_ s)=\max\rho(C)\) where the maximum is taken over all systems \(C\) of the noted form. It was previously shown that if the field of scalars is infinite then \(R(V;m_ 1,\ldots,m_ s)=m+1\) and it thus suffices to consider \(V\) over a finite field. The relationship of this problem with coding theory is also considered.

MSC:

11T99 Finite fields and commutative rings (number-theoretic aspects)
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
94B40 Arithmetic codes
15B33 Matrices over special rings (quaternions, finite fields, etc.)
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References:

[1] Berlekamp, E. R., Algebraic Coding Theory (1968), McGraw-Hill: McGraw-Hill New York · Zbl 0199.54101
[2] Lidl, R.; Niederreiter, H., Finite Fields (1983), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0554.12010
[3] MacWilliams, F. J.; Sloane, N. J.A., The Theory of Error-Correcting Codes (1977), North-Holland: North-Holland Amsterdam · Zbl 0369.94008
[4] McEliece, R. J., The Theory of Information and Coding (1977), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0376.94011
[5] Niederreiter, H., Point sets and sequences with small discrepancy, Monatsh. Math., 104, 273-337 (1987) · Zbl 0626.10045
[6] van Lint, J. H., Introduction to Coding Theory (1982), Springer: Springer Berlin · Zbl 0485.94015
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