Best Known (108−19, 108, s)-Nets in Base 5
(108−19, 108, 8682)-Net over F5 — Constructive and digital
Digital (89, 108, 8682)-net over F5, using
- 51 times duplication [i] based on digital (88, 107, 8682)-net over F5, using
- net defined by OOA [i] based on linear OOA(5107, 8682, F5, 19, 19) (dual of [(8682, 19), 164851, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5107, 78139, F5, 19) (dual of [78139, 78032, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(5107, 78140, F5, 19) (dual of [78140, 78033, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [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(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(51, 15, F5, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(5107, 78140, F5, 19) (dual of [78140, 78033, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5107, 78139, F5, 19) (dual of [78139, 78032, 20]-code), using
- net defined by OOA [i] based on linear OOA(5107, 8682, F5, 19, 19) (dual of [(8682, 19), 164851, 20]-NRT-code), using
(108−19, 108, 44996)-Net over F5 — Digital
Digital (89, 108, 44996)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5108, 44996, F5, 19) (dual of [44996, 44888, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(5108, 78142, F5, 19) (dual of [78142, 78034, 20]-code), using
- construction XX applied to Ce(18) ⊂ Ce(16) ⊂ 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(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(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(51, 16, F5, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(18) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(5108, 78142, F5, 19) (dual of [78142, 78034, 20]-code), using
(108−19, 108, large)-Net in Base 5 — Upper bound on s
There is no (89, 108, large)-net in base 5, because
- 17 times m-reduction [i] would yield (89, 91, large)-net in base 5, but