Best Known (126−17, 126, s)-Nets in Base 5
(126−17, 126, 244147)-Net over F5 — Constructive and digital
Digital (109, 126, 244147)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 8, 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 (101, 118, 244141)-net over F5, using
- net defined by OOA [i] based on linear OOA(5118, 244141, F5, 17, 17) (dual of [(244141, 17), 4150279, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(5118, 1953129, F5, 17) (dual of [1953129, 1953011, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(5118, 1953134, F5, 17) (dual of [1953134, 1953016, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(5118, 1953125, F5, 17) (dual of [1953125, 1953007, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(5109, 1953125, F5, 16) (dual of [1953125, 1953016, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(50, 9, F5, 0) (dual of [9, 9, 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(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(5118, 1953134, F5, 17) (dual of [1953134, 1953016, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(5118, 1953129, F5, 17) (dual of [1953129, 1953011, 18]-code), using
- net defined by OOA [i] based on linear OOA(5118, 244141, F5, 17, 17) (dual of [(244141, 17), 4150279, 18]-NRT-code), using
- digital (0, 8, 6)-net over F5, using
(126−17, 126, 1072636)-Net over F5 — Digital
Digital (109, 126, 1072636)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5126, 1072636, F5, 17) (dual of [1072636, 1072510, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(5126, 1953134, F5, 17) (dual of [1953134, 1953008, 18]-code), using
- (u, u+v)-construction [i] based on
- linear OA(58, 9, F5, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,5)), using
- dual of repetition code with length 9 [i]
- linear OA(5118, 1953125, F5, 17) (dual of [1953125, 1953007, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(58, 9, F5, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,5)), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(5126, 1953134, F5, 17) (dual of [1953134, 1953008, 18]-code), using
(126−17, 126, large)-Net in Base 5 — Upper bound on s
There is no (109, 126, large)-net in base 5, because
- 15 times m-reduction [i] would yield (109, 111, large)-net in base 5, but