Table 5 Some known QEC MDS codes with minimum distance \(d>\frac{q+1}{2}\).
Code length | Constraints | Minimum distances | References |
|---|---|---|---|
\(\frac{q^2-1}{2}\) | q odd | \(2\le d\le q\) | |
\(\lambda (q+1)\) | \(\lambda\) odd, \(\lambda \mid (q-1)\) | \(2 \le d \le \frac{q+1}{2}+\lambda\) | |
\(2\lambda (q+1)\) | \(q \equiv 1 (\textrm{mod}~4)\), \(\lambda\) odd, \(\lambda \mid (q-1)\) | \(2 \le d \le \frac{q+1}{2}+2 \lambda\) | |
\(\lambda (q-1)\) | \(\lambda = \frac{q+1 }{r}\) with even r | \(2 \le d \le \frac{q+1}{2}+ \lambda -1\) | |
\(\lambda (q-1)\) | \(\lambda = \frac{q+1 }{r}\) with odd r | \(2 \le d \le \frac{q+1}{2}+ \frac{\lambda }{2}-1\) | |
\(\frac{q^2-1}{h}\) | q odd, \(h \mid (q+1)\), \(h \in {{3,5,7}}\) | \(2 \le d \le \frac{(q+1)(h+1)}{2h}-1\) | |
\(2t(q-1)\) | \(8 \mid (q+1)\), \(t \mid (q+1)\) | \(2 \le d \le 6t+1\) | |
\(3(q-1)\) | \(3^{2} \mid (q+1)\) with odd q | \(2 \le d \le \frac{q+5}{2}\) | |
\((2t+1)\frac{q^{2}-1}{2s+1}\) | \((2s+1) \mid (q+1)\), \(1 \le t \le s\) | \(2 \le d \le (s+t+1)\frac{q-1}{2s+1}-1\) | |
\(2t \frac{q^{2}-1}{2s+1}\) | \((2s+1) \mid (q+1)\), \(1 \le t \le s-1\) | \(2 \le d \le (s+t+1)\frac{q-1}{2s+1}-1\) | |
\(2t \frac{q^{2}-1}{2s}\) | \(2s \mid (q+1)\), \(1 \le t \le s\) | \(2 \le d \le (s+t) \frac{q-1}{2s}-1\) | |
\(r \frac{q^{2}-1}{2s}\) | \(2s \mid (q+1)\), \(1 \le r \le 2s\) | \(2 \le d \le (s+1) \frac{q-1}{2s}\) | |
\(r \frac{q^{2}-1}{s}\) | \(s \mid (q-1)\), \(1 \le r \le s\) | \(2 \le d \le r \frac{q-1}{s}\) | |
\((2t+1)\frac{q^{2}-1}{2s}\) | \(2s \mid (q-1)\), \(1 \le t \le 2s\) | \(2 \le d \le (s+t) \frac{q-1}{2s}-1\) | |
\(\frac{q^2-1}{4}+\frac{q^2-1}{h}\) | \(\frac{2(q-1)}{h}=2\tau +1\) | \(2\le d \le \frac{q-1}{2}+\tau\) | |
\(\frac{q^2-1}{4}+\frac{2(q^2-1)}{h}\) | \(\frac{2(q-1)}{h}=2\tau +1\), \((h\ne 4)\) | \(2\le d \le \frac{q-1}{2}+2\tau +1\) | |
\(m(q-1)\) | \(1 \le m \le q\) | \(2 \le d \le \lfloor \frac{mq-1}{q+1}\rfloor +1\) | |
\(\frac{2(r_1+1)r_2(q^2-1)}{h}\) | \(q\equiv 3~(\text {mod}~4)\), \(h=h_1h_2\ge 9\), \(h_1\mid \frac{q-1}{2}\), \(h_2\mid (q+1)\), \(r_1\le h_1-1\), odd \(h_2>2r_2\) | \(2 \le d \le \text {max}\{\frac{(r_1-1)(q-1)}{2h_1},\) \(\frac{2(r_2-1)(q+1)}{h_2}\}+2\) | |
\(\frac{(5t+q-1)(q+1)}{9}\) | \(3\mid (q+1)\),\(t\mid (q+1)\),\(3\mid (t-1)\) | \(2 \le d \le \frac{q+2t-1}{3} +1\) | |
\(\frac{(7t+q-1)(q+1)}{9}\) | \(3\mid (q+1)\),\(t\mid (q+1)\),\(3\mid (t+1)\) | \(2 \le d \le \frac{q+1}{3}+t\) | |
\(s(q+1)\) | \(1 \le s \le q-1\) | \(2 \le d \le s\) | |
\((\mu +1)\frac{q^2-1}{\gamma }\) | \(q\equiv -1~(\text {mod}~4)\), \(\gamma \mid 2(q-1)\) and \(\gamma \nmid (q-1)\), \(0\le \mu \le \frac{\gamma }{4}-1\) | \(2\le d \le (\frac{\gamma +4}{8}+\mu )\frac{2(q-1)}{\gamma }+1\) | Theorem 8 |
\((2\mu +1)\frac{q^2-1}{\gamma }\) | \(\gamma \equiv 2(\text {mod 4})\), \(\gamma \mid (q+1)\), \(0\le \mu \le \frac{\gamma -2}{4}\) | \(2 \le d \le \frac{q+1}{2}+(\mu +1)\frac{q+1}{\gamma }-1\) | Theorem 12 |
