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A construction of two-weight codes and its applications. (English) Zbl 1370.94590
Summary: It is well-known that there exists a constant-weight $$[s \theta_{k-1},k, sq^{k-1}]_q$$ code for any positive integer $$s$$, which is an $$s$$-fold simplex code, where $$\theta_{j}=(q^{j+1}-1)/(q-1)$$. This gives an upper bound $$n_q(k, sq^{k-1}+d) \leq s \theta_{k-1} + n_q(k,d)$$ for any positive integer $$d$$, where $$n_q(k,d)$$ is the minimum length $$n$$ for which an $$[n,k,d]_q$$ code exists. We construct a two-weight $$[s \theta_{k-1}+1,k, sq^{k-1}]_q$$ code for $$1 \leq s \leq k-3$$, which gives a better upper bound $$n_q(k, s q^{k-1}+d) \leq s \theta_{k-1} +1 + n_q(k-1,d)$$ for $$1 \leq d \leq q^s$$. As another application, we prove that $$n_q(5,d)=\sum_{i=0}^{4}{\lceil d/q^i \rceil}$$ for $$q^{4}+1 \leq d \leq q^4+q$$ for any prime power $$q$$.

##### MSC:
 94B27 Geometric methods (including applications of algebraic geometry) applied to coding theory 94B05 Linear codes, general 51E20 Combinatorial structures in finite projective spaces 05B25 Combinatorial aspects of finite geometries
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