Best Known (93, 93+18, s)-Nets in Base 5
(93, 93+18, 8697)-Net over F5 — Constructive and digital
Digital (93, 111, 8697)-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 (81, 99, 8681)-net over F5, using
- net defined by OOA [i] based on linear OOA(599, 8681, F5, 18, 18) (dual of [(8681, 18), 156159, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(599, 78129, F5, 18) (dual of [78129, 78030, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(599, 78132, F5, 18) (dual of [78132, 78033, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(599, 78125, F5, 18) (dual of [78125, 78026, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(592, 78125, F5, 17) (dual of [78125, 78033, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(599, 78132, F5, 18) (dual of [78132, 78033, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(599, 78129, F5, 18) (dual of [78129, 78030, 19]-code), using
- net defined by OOA [i] based on linear OOA(599, 8681, F5, 18, 18) (dual of [(8681, 18), 156159, 19]-NRT-code), using
- digital (3, 12, 16)-net over F5, using
(93, 93+18, 78179)-Net over F5 — Digital
Digital (93, 111, 78179)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5111, 78179, F5, 18) (dual of [78179, 78068, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(10) [i] based on
- linear OA(599, 78125, F5, 18) (dual of [78125, 78026, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(512, 54, F5, 6) (dual of [54, 42, 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(17) ⊂ Ce(10) [i] based on
(93, 93+18, large)-Net in Base 5 — Upper bound on s
There is no (93, 111, large)-net in base 5, because
- 16 times m-reduction [i] would yield (93, 95, large)-net in base 5, but