Best Known (126, 126+22, s)-Nets in Base 5
(126, 126+22, 35518)-Net over F5 — Constructive and digital
Digital (126, 148, 35518)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 11, 6)-net over F5, using
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 0 and N(F) ≥ 6, using
- the rational function field F5(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- digital (115, 137, 35512)-net over F5, using
- net defined by OOA [i] based on linear OOA(5137, 35512, F5, 22, 22) (dual of [(35512, 22), 781127, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(5137, 390632, F5, 22) (dual of [390632, 390495, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(5137, 390633, F5, 22) (dual of [390633, 390496, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(5137, 390625, F5, 22) (dual of [390625, 390488, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(5129, 390625, F5, 21) (dual of [390625, 390496, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(50, 8, F5, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(5137, 390633, F5, 22) (dual of [390633, 390496, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(5137, 390632, F5, 22) (dual of [390632, 390495, 23]-code), using
- net defined by OOA [i] based on linear OOA(5137, 35512, F5, 22, 22) (dual of [(35512, 22), 781127, 23]-NRT-code), using
- digital (0, 11, 6)-net over F5, using
(126, 126+22, 284875)-Net over F5 — Digital
Digital (126, 148, 284875)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5148, 284875, F5, 22) (dual of [284875, 284727, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(5148, 390637, F5, 22) (dual of [390637, 390489, 23]-code), using
- (u, u+v)-construction [i] based on
- linear OA(511, 12, F5, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,5)), using
- dual of repetition code with length 12 [i]
- linear OA(5137, 390625, F5, 22) (dual of [390625, 390488, 23]-code), using
- an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(511, 12, F5, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,5)), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(5148, 390637, F5, 22) (dual of [390637, 390489, 23]-code), using
(126, 126+22, large)-Net in Base 5 — Upper bound on s
There is no (126, 148, large)-net in base 5, because
- 20 times m-reduction [i] would yield (126, 128, large)-net in base 5, but