Best Known (111−19, 111, s)-Nets in Base 5
(111−19, 111, 8683)-Net over F5 — Constructive and digital
Digital (92, 111, 8683)-net over F5, using
- 52 times duplication [i] based on digital (90, 109, 8683)-net over F5, using
- net defined by OOA [i] based on linear OOA(5109, 8683, F5, 19, 19) (dual of [(8683, 19), 164868, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5109, 78148, F5, 19) (dual of [78148, 78039, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(5109, 78149, F5, 19) (dual of [78149, 78040, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(5106, 78125, F5, 19) (dual of [78125, 78019, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(585, 78125, F5, 16) (dual of [78125, 78040, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 24 = 52−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(5109, 78149, F5, 19) (dual of [78149, 78040, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5109, 78148, F5, 19) (dual of [78148, 78039, 20]-code), using
- net defined by OOA [i] based on linear OOA(5109, 8683, F5, 19, 19) (dual of [(8683, 19), 164868, 20]-NRT-code), using
(111−19, 111, 59779)-Net over F5 — Digital
Digital (92, 111, 59779)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5111, 59779, F5, 19) (dual of [59779, 59668, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(5111, 78152, F5, 19) (dual of [78152, 78041, 20]-code), using
- construction XX applied to Ce(18) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- linear OA(5106, 78125, F5, 19) (dual of [78125, 78019, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(585, 78125, F5, 16) (dual of [78125, 78040, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(578, 78125, F5, 14) (dual of [78125, 78047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(53, 25, F5, 2) (dual of [25, 22, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 31, F5, 2) (dual of [31, 28, 3]-code), using
- Hamming code H(3,5) [i]
- discarding factors / shortening the dual code based on linear OA(53, 31, F5, 2) (dual of [31, 28, 3]-code), using
- linear OA(51, 2, F5, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(18) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(5111, 78152, F5, 19) (dual of [78152, 78041, 20]-code), using
(111−19, 111, large)-Net in Base 5 — Upper bound on s
There is no (92, 111, large)-net in base 5, because
- 17 times m-reduction [i] would yield (92, 94, large)-net in base 5, but