Best Known (54−11, 54, s)-Nets in Base 5
(54−11, 54, 3131)-Net over F5 — Constructive and digital
Digital (43, 54, 3131)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 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 (38, 49, 3125)-net over F5, using
- net defined by OOA [i] based on linear OOA(549, 3125, F5, 11, 11) (dual of [(3125, 11), 34326, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(549, 15626, F5, 11) (dual of [15626, 15577, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- OOA 5-folding and stacking with additional row [i] based on linear OA(549, 15626, F5, 11) (dual of [15626, 15577, 12]-code), using
- net defined by OOA [i] based on linear OOA(549, 3125, F5, 11, 11) (dual of [(3125, 11), 34326, 12]-NRT-code), using
- digital (0, 5, 6)-net over F5, using
(54−11, 54, 13541)-Net over F5 — Digital
Digital (43, 54, 13541)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(554, 13541, F5, 11) (dual of [13541, 13487, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(554, 15632, F5, 11) (dual of [15632, 15578, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([1,5]) [i] based on
- linear OA(549, 15626, F5, 11) (dual of [15626, 15577, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(548, 15626, F5, 5) (dual of [15626, 15578, 6]-code), using the narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(55, 6, F5, 5) (dual of [6, 1, 6]-code or 6-arc in PG(4,5)), using
- dual of repetition code with length 6 [i]
- construction X applied to C([0,5]) ⊂ C([1,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(554, 15632, F5, 11) (dual of [15632, 15578, 12]-code), using
(54−11, 54, large)-Net in Base 5 — Upper bound on s
There is no (43, 54, large)-net in base 5, because
- 9 times m-reduction [i] would yield (43, 45, large)-net in base 5, but