Best Known (111, 135, s)-Nets in Base 5
(111, 135, 6511)-Net over F5 — Constructive and digital
Digital (111, 135, 6511)-net over F5, using
- 51 times duplication [i] based on digital (110, 134, 6511)-net over F5, using
- net defined by OOA [i] based on linear OOA(5134, 6511, F5, 24, 24) (dual of [(6511, 24), 156130, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(5134, 78132, F5, 24) (dual of [78132, 77998, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(22) [i] based on
- linear OA(5134, 78125, F5, 24) (dual of [78125, 77991, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(5127, 78125, F5, 23) (dual of [78125, 77998, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(23) ⊂ Ce(22) [i] based on
- OA 12-folding and stacking [i] based on linear OA(5134, 78132, F5, 24) (dual of [78132, 77998, 25]-code), using
- net defined by OOA [i] based on linear OOA(5134, 6511, F5, 24, 24) (dual of [(6511, 24), 156130, 25]-NRT-code), using
(111, 135, 40925)-Net over F5 — Digital
Digital (111, 135, 40925)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5135, 40925, F5, 24) (dual of [40925, 40790, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(5135, 78140, F5, 24) (dual of [78140, 78005, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- linear OA(5134, 78125, F5, 24) (dual of [78125, 77991, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(5120, 78125, F5, 22) (dual of [78125, 78005, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(23) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(5135, 78140, F5, 24) (dual of [78140, 78005, 25]-code), using
(111, 135, large)-Net in Base 5 — Upper bound on s
There is no (111, 135, large)-net in base 5, because
- 22 times m-reduction [i] would yield (111, 113, large)-net in base 5, but