# The effect of channel error on the determinant of transmitted matrix

Assume the following matrix

$$E:=\left( \begin{array}{ccccc} e_1 & e_2 & \cdots & e_{p-1} & e_{p}\\ e_{p+1} & e_{p+2} & \cdots & e_{2p-1} & e_{2p} \\ \vdots & \vdots & \vdots & \vdots &\vdots \\ e_{p^2-p+1} & e_{p^2-p+2} & \cdots & e_{p^2-1} & e_{p^2} \end{array} \right)_{p\times p}=(e_{i})\in {\Bbb{R}}^{p \times p}\, .$$ Let the determinant of the matrix $E$ is a number like $m$, and suppose $l_i$, $1\leq i \leq p^2$, is the number of decimal digits in the $i$th entry of the matrix $E$. Imagine, we are as a sender in a communication system and we want to send the entries of the matrix $E$, through a noisy channel. Suppose the channel error probability per decimal symbol is $q$. We denote the received matrix $E$, by $\widehat{E}$. Consider, the entries of the matrix $\widehat{E}$, are received with $k$ errors, where $2\leq k \leq p^2$. I mean, we have at least two errors. Now, my question it is, with which probability, the determinant of the received matrix $\widehat{E}$ is equal the $m$, when $k$ entries of the matrix $\widehat{E}$, are received with errors. Notice that, $k$ errors can be happen in the ${{p^2}\choose{k}}=\frac{p^2!}{k!\, (p^2-k)!}$ cases in the matrix $\widehat{E}$.

I would greatly appreciate for any suggestions.

• This question is one of the open problems in the decoding process in a new class of coding theory called Fibonacci coding. For more details, please see the following article: dx.doi.org/10.1016/j.chaos.2005.12.054 – Amin235 Sep 6 '16 at 5:16
• This question needs more background/motivation. The abstract of the paper you point to is quite horribly written. The claim of "93.33% correctability" sounds like snake oil, given there is supposedly no redundancy. So you are violating the basic principles of information and coding theory and correcting a vast majority of errors, how exactly? – kodlu Sep 10 '16 at 1:54
• @kodlu you right. The article that i mentioned, is not suitable for learning the fibonacci coding. One of the best source to familiar with code rate and error-correcting capability of Fibonacci coding is the last chapter of the following thesis: docdro.id/EZSVY8i – Amin235 Sep 10 '16 at 12:23