Input-Output Distance Properties of Good Linear Codes

Consider a linear code defined as a mapping between vector spaces of dimensions <tex>$k$</tex> and <tex>$n$</tex>. Let <tex>$\beta^{\ast}$</tex> denote the minimal (relative) weight among all images of input vectors of full Hamming weight <tex>$k$</tex>. Operationally, <tex>$\beta^{\ast}$</tex> characterizes the threshold for adversarial (erasure) noise beyond which decoder is guaranteed to produce estimate of k-input with 100% symbol error rate (SER). This paper studies the relation between <tex>$\beta^{\ast}$</tex> and <tex>$\delta$</tex>, the minimum distance of the code, which gives the threshold for 0 % SER. An optimal tradeoff between <tex>$\beta^{\ast}$</tex> and <tex>$\delta$</tex> is obtained (over large alphabets) and all linear codes achieving <tex>$\beta^{\ast}=1$</tex> are classified: they are repetition-like. More generally, a design criteria is proposed for codes with favorable graceful degradation properties. As an example, it is shown that in an overdetermined system of <tex>$n$</tex> homogeneous linear equations in <tex>$k$</tex> variables (over a field) it is always possible to satisfy some <tex>$k-1$</tex> equations with non-zero assignments to every unknown, provided that any subset of <tex>$k$</tex> equations is linearly independent. This statement is true if and only if <tex>$n\geq 2k-1$</tex>.