Best Known (78−10, 78, s)-Nets in Base 5
(78−10, 78, 390635)-Net over F5 — Constructive and digital
Digital (68, 78, 390635)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 6, 10)-net over F5, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 1 and N(F) ≥ 10, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- digital (62, 72, 390625)-net over F5, using
- net defined by OOA [i] based on linear OOA(572, 390625, F5, 10, 10) (dual of [(390625, 10), 3906178, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(572, 1953125, F5, 10) (dual of [1953125, 1953053, 11]-code), using
- 1 times truncation [i] based on linear OA(573, 1953126, F5, 11) (dual of [1953126, 1953053, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1953126 | 518−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(573, 1953126, F5, 11) (dual of [1953126, 1953053, 12]-code), using
- OA 5-folding and stacking [i] based on linear OA(572, 1953125, F5, 10) (dual of [1953125, 1953053, 11]-code), using
- net defined by OOA [i] based on linear OOA(572, 390625, F5, 10, 10) (dual of [(390625, 10), 3906178, 11]-NRT-code), using
- digital (1, 6, 10)-net over F5, using
(78−10, 78, 1953166)-Net over F5 — Digital
Digital (68, 78, 1953166)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(578, 1953166, F5, 10) (dual of [1953166, 1953088, 11]-code), using
- construction X applied to Ce(10) ⊂ Ce(5) [i] based on
- linear OA(573, 1953125, F5, 11) (dual of [1953125, 1953052, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(537, 1953125, F5, 6) (dual of [1953125, 1953088, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(55, 41, F5, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,5)), using
- discarding factors / shortening the dual code based on linear OA(55, 42, F5, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,5)), using
- construction X applied to Ce(10) ⊂ Ce(5) [i] based on
(78−10, 78, large)-Net in Base 5 — Upper bound on s
There is no (68, 78, large)-net in base 5, because
- 8 times m-reduction [i] would yield (68, 70, large)-net in base 5, but