Best Known (114, 114+19, s)-Nets in Base 5
(114, 114+19, 43419)-Net over F5 — Constructive and digital
Digital (114, 133, 43419)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (3, 12, 16)-net over F5, using
- net from sequence [i] based on digital (3, 15)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 3 and N(F) ≥ 16, using
- net from sequence [i] based on digital (3, 15)-sequence over F5, using
- digital (102, 121, 43403)-net over F5, using
- net defined by OOA [i] based on linear OOA(5121, 43403, F5, 19, 19) (dual of [(43403, 19), 824536, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5121, 390628, F5, 19) (dual of [390628, 390507, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(5121, 390633, F5, 19) (dual of [390633, 390512, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(5121, 390625, F5, 19) (dual of [390625, 390504, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(5113, 390625, F5, 18) (dual of [390625, 390512, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(5121, 390633, F5, 19) (dual of [390633, 390512, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5121, 390628, F5, 19) (dual of [390628, 390507, 20]-code), using
- net defined by OOA [i] based on linear OOA(5121, 43403, F5, 19, 19) (dual of [(43403, 19), 824536, 20]-NRT-code), using
- digital (3, 12, 16)-net over F5, using
(114, 114+19, 390685)-Net over F5 — Digital
Digital (114, 133, 390685)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5133, 390685, F5, 19) (dual of [390685, 390552, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(11) [i] based on
- linear OA(5121, 390625, F5, 19) (dual of [390625, 390504, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(573, 390625, F5, 12) (dual of [390625, 390552, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(512, 60, F5, 6) (dual of [60, 48, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(512, 62, F5, 6) (dual of [62, 50, 7]-code), using
- the cyclic code C(A) with length 62 | 53−1, defining set A = {4,8,11,17}, and minimum distance d ≥ |{8,11,14,…,23}|+1 = 7 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(512, 62, F5, 6) (dual of [62, 50, 7]-code), using
- construction X applied to Ce(18) ⊂ Ce(11) [i] based on
(114, 114+19, large)-Net in Base 5 — Upper bound on s
There is no (114, 133, large)-net in base 5, because
- 17 times m-reduction [i] would yield (114, 116, large)-net in base 5, but