Best Known (58, 58+12, s)-Nets in Base 5
(58, 58+12, 13028)-Net over F5 — Constructive and digital
Digital (58, 70, 13028)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 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 (52, 64, 13022)-net over F5, using
- net defined by OOA [i] based on linear OOA(564, 13022, F5, 12, 12) (dual of [(13022, 12), 156200, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(564, 78132, F5, 12) (dual of [78132, 78068, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(564, 78125, F5, 12) (dual of [78125, 78061, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(557, 78125, F5, 11) (dual of [78125, 78068, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(50, 7, F5, 0) (dual of [7, 7, 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(11) ⊂ Ce(10) [i] based on
- OA 6-folding and stacking [i] based on linear OA(564, 78132, F5, 12) (dual of [78132, 78068, 13]-code), using
- net defined by OOA [i] based on linear OOA(564, 13022, F5, 12, 12) (dual of [(13022, 12), 156200, 13]-NRT-code), using
- digital (0, 6, 6)-net over F5, using
(58, 58+12, 75296)-Net over F5 — Digital
Digital (58, 70, 75296)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(570, 75296, F5, 12) (dual of [75296, 75226, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(570, 78132, F5, 12) (dual of [78132, 78062, 13]-code), using
- (u, u+v)-construction [i] based on
- linear OA(56, 7, F5, 6) (dual of [7, 1, 7]-code or 7-arc in PG(5,5)), using
- dual of repetition code with length 7 [i]
- linear OA(564, 78125, F5, 12) (dual of [78125, 78061, 13]-code), using
- an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(56, 7, F5, 6) (dual of [7, 1, 7]-code or 7-arc in PG(5,5)), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(570, 78132, F5, 12) (dual of [78132, 78062, 13]-code), using
(58, 58+12, large)-Net in Base 5 — Upper bound on s
There is no (58, 70, large)-net in base 5, because
- 10 times m-reduction [i] would yield (58, 60, large)-net in base 5, but