Article Contents
Article Contents

# The weight distribution of quasi-quadratic residue codes

The first author is supported by Simons Foundation grant MSN179747

• We investigate a family of codes called quasi-quadratic residue (QQR) codes. We are interested in these codes mainly for two reasons: Firstly, they have close relations with hyperelliptic curves and Goppa's Conjecture, and serve as a strong tool in studying those objects. Secondly, they are very good codes. Computational results show they have large minimum distances when $p\equiv 3 \pmod 8$.

Our studies focus on the weight distributions of these codes. We will prove a new discovery about their weight polynomials, i.e. they are divisible by $(x^2 + y^2)^{d-1}$, where $d$ is the corresponding minimum distance. We also show that the weight distributions of these codes are asymptotically normal. Based on the relation between QQR codes and hyperelliptic curves, we will also prove a result on the point distribution on hyperelliptic curves.

Mathematics Subject Classification: Primary: 94B15, 94B60; Secondary: 11G20.

 Citation:

• Figure 1.  distribution comparison

Table 1.  Computational Results

 $p$ $d$ $\delta$ Divisible by 3 2 0.33 $(x^2 + y^2)^3$ 11 6 0.27 $(x^2 + y^2)^7$ 19 8 0.21 $(x^2 + y^2)^7$ 43 14 0.16 $(x^2 + y^2)^{15}$ 59 18 0.15 $(x^2 + y^2)^{19}$ 67 22 0.16 $(x^2 + y^2)^{23}$

Table 2.  Weight polynomials posted on [18]

 $p$ $k$ $d$ Divisible by 89 45 17 $(x+y)^{17}$ 97 49 15 $(x+y)^{15}$ 103 52 19 $(x+y)^{19}$ 113 57 15 $(x+y)$ 127 64 19 $(x+y)$ 137 69 21 $(x+y)$ 151 76 19 $(x+y)$ 167 84 23 $(x+y)$

Table 3.  Weight polynomials in references

 $p$ $k$ $d$ Divisible by 137 69 21 $(x+y)^{21}$ 151 76 19 $(x+y)^{19}$ 167 84 23 $(x+y)^{23}$

Table 4.  Correction for $p = 127$

 $i$ $A_i$ in table $A_i$ corrected 51 223367511592873280 223367511592873284 52 326460209251122496 326460209251122492 55 840260234424082176 840260234424082220 56 1080334587116677120 1080334587116677140 59 1899366974583683328 1899366974583683220 60 2152615904528174336 2152615904528174316 63 2596788489999036416 2596788489999036307
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