Best Known (127−19, 127, s)-Nets in Base 5
(127−19, 127, 43406)-Net over F5 — Constructive and digital
Digital (108, 127, 43406)-net over F5, using
- 51 times duplication [i] based on digital (107, 126, 43406)-net over F5, using
- net defined by OOA [i] based on linear OOA(5126, 43406, F5, 19, 19) (dual of [(43406, 19), 824588, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5126, 390655, F5, 19) (dual of [390655, 390529, 20]-code), using
- construction XX applied to Ce(18) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- linear OA(5121, 390625, F5, 19) (dual of [390625, 390504, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(597, 390625, F5, 16) (dual of [390625, 390528, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(589, 390625, F5, 14) (dual of [390625, 390536, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(53, 28, F5, 2) (dual of [28, 25, 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
- OOA 9-folding and stacking with additional row [i] based on linear OA(5126, 390655, F5, 19) (dual of [390655, 390529, 20]-code), using
- net defined by OOA [i] based on linear OOA(5126, 43406, F5, 19, 19) (dual of [(43406, 19), 824588, 20]-NRT-code), using
(127−19, 127, 271937)-Net over F5 — Digital
Digital (108, 127, 271937)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5127, 271937, F5, 19) (dual of [271937, 271810, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(5127, 390657, F5, 19) (dual of [390657, 390530, 20]-code), using
- construction XX applied to Ce(18) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- linear OA(5121, 390625, F5, 19) (dual of [390625, 390504, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(597, 390625, F5, 16) (dual of [390625, 390528, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(589, 390625, F5, 14) (dual of [390625, 390536, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(53, 29, F5, 2) (dual of [29, 26, 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, 3, F5, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, 5, F5, 1) (dual of [5, 4, 2]-code), using
- Reed–Solomon code RS(4,5) [i]
- discarding factors / shortening the dual code based on linear OA(51, 5, F5, 1) (dual of [5, 4, 2]-code), using
- construction XX applied to Ce(18) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(5127, 390657, F5, 19) (dual of [390657, 390530, 20]-code), using
(127−19, 127, large)-Net in Base 5 — Upper bound on s
There is no (108, 127, large)-net in base 5, because
- 17 times m-reduction [i] would yield (108, 110, large)-net in base 5, but