Best Known (51, 51+8, s)-Nets in Base 5
(51, 51+8, 488289)-Net over F5 — Constructive and digital
Digital (51, 59, 488289)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 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 (47, 55, 488283)-net over F5, using
- net defined by OOA [i] based on linear OOA(555, 488283, F5, 8, 8) (dual of [(488283, 8), 3906209, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(555, 1953132, F5, 8) (dual of [1953132, 1953077, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(555, 1953134, F5, 8) (dual of [1953134, 1953079, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(555, 1953125, F5, 8) (dual of [1953125, 1953070, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(546, 1953125, F5, 7) (dual of [1953125, 1953079, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(555, 1953134, F5, 8) (dual of [1953134, 1953079, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(555, 1953132, F5, 8) (dual of [1953132, 1953077, 9]-code), using
- net defined by OOA [i] based on linear OOA(555, 488283, F5, 8, 8) (dual of [(488283, 8), 3906209, 9]-NRT-code), using
- digital (0, 4, 6)-net over F5, using
(51, 51+8, 1953151)-Net over F5 — Digital
Digital (51, 59, 1953151)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(559, 1953151, F5, 8) (dual of [1953151, 1953092, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(3) [i] based on
- linear OA(555, 1953125, F5, 8) (dual of [1953125, 1953070, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(528, 1953125, F5, 4) (dual of [1953125, 1953097, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(54, 26, F5, 3) (dual of [26, 22, 4]-code or 26-cap in PG(3,5)), using
- construction X applied to Ce(7) ⊂ Ce(3) [i] based on
(51, 51+8, large)-Net in Base 5 — Upper bound on s
There is no (51, 59, large)-net in base 5, because
- 6 times m-reduction [i] would yield (51, 53, large)-net in base 5, but