Abstract

Expansions in a finite frame are considered as a continuous linear redundant coding. It is shown that coding of an element from an \(N\)-dimensional space with a frame consisting of \((N+M)\) elements allows detecting up to \(M\) errors and correcting up to \(\left\lfloor\dfrac{M}{2}\right\rfloor\) errors. It is also pointed out that these results are sharp. The results are direct continuous analogs of the classical statements from the discrete coding theory.

中文翻译：

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作为连续冗余码的帧

摘要

有限帧中的扩展被视为连续线性冗余编码。结果表明，使用由\((N+M)\) 个元素组成的框架对\(N\)维空间中的元素进行编码允许检测多达\(M\) 个错误并纠正多达\(\左\lfloor\dfrac{M}{2}\right\rfloor\)错误。还指出这些结果是尖锐的。结果是离散编码理论中经典陈述的直接连续模拟。