On deep holes of primitive projective Reed-Solomon codes
Xu Xiaofan, Hong Shaofang, Yongchao Xu · Scientia Sinica Mathematica · 2018
Primitive projectiveReed-Solomon code, which is an important class ofmaximum distance separable codes, is currently widely usedin digital communication. We usually usethe maximum likelihood decoding algorithm in thedecoding process of Reed-Solomon codes. For the receivedword ${\boldsymbol~u}\in\mathbb{F}_q^n$, the maximum likelihooddecoding algorithm lies in determining its error distance$d({\boldsymbol~u},C)$. It is well known that $d~({\boldsymbol~u},~C)\leq~\rho(C)$with $\rho(C)$ being the covering radius of code $C$.If $d({\boldsymbol~u},~C)=\rho(C)$, then ${\boldsymbol~u}$ is calleda deep hole of $C$. In this paper, we obtain a class of deepholes of primitive projective Reed-Solomon codes${\rm~PPRS}_q(\mathbb{F}_q^*,k)$. In fact, by usingthe generating matrix of maximaldistance separable codes over the finite field $\mathbb{F}_q$and Vandermonde determinant, we show that if $q\ge~4$, $k$ is aninteger with $2\le~k\le~q-2$ and the Lagrange interpolating polynomialof the first $q-1$ components of the received word ${\boldsymbol~u}$ is$\lambda~x^{q-2}+f_{\le{k-2}}(x)$, with $\lambda\in\mathbb{F}_q^*$and $f_{\leqslant{k-2}}(x)$ being a polynomial of degree no morethan $k-2$ over $\mathbb{F}_q$, and the $q$-th component of${\boldsymbol~u}$ is $0$, then ${\boldsymbol~u}$ is a deep hole of the primitiveprojective Reed-Solomon code ${\rm~PPRS}_q(\mathbb{F}_q^*,k)$.