## 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.)

### Keywords:

vector spaces; finite fields; linear dependence; coding theory
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### References:

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