**Authors:** L. Rabefihavanana, H. Andriatahiny, T. Rabeherimanana

**Keywords:** Error correction code, encoding, decoding, sub-exceeding function.

### Abstract

In this paper, we present linear systematic error-correcting codes $\mathcal{L}_{k}$ and $\mathcal{L}^{+}_{k}$ which are the results of our research on the sub-exceeding functions. Given an integer $ k $ such that $ k \geq 3$, these two codes are respectively $[2k,k]$ and $[3k,k]$ linear codes. The minimum distance of $\mathcal{L}_3$ is 3 and for $k\geq 4$ the minimum distance of $\mathcal{L}_k$ is 4. The code $\mathcal{L}_k ^{+}$, the minimum distances are respectively 5 and 6 for $ k = 4 $ and $ k \geq 5 $. By calculating the complexity of the algorithms, our codes have fast and efficient decoding. Then, for a short and medium distance data transmission (wifi network, bluetooth, cable, ...), we see that the codes mentioned above present many advantages.

Luc Rabefihavanana

Department of Mathematics and Computer Science

Faculty of Sciences, University of Antananarivo, Madagascar

Phone: (+261) 34 74 877 40

E-mail:

Harinaivo Andriatahiny

Department of Mathematics and Computer Science

Faculty of Sciences, University of Antananarivo, Madagascar

E-mail:

Toussaint Joseph Rabeherimanana

Department of Mathematics and Computer Science

Faculty of Sciences, University of Antananarivo, Madagascar

E-mail:

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