Best Known (109, 109+19, s)-Nets in Base 5
(109, 109+19, 43407)-Net over F5 — Constructive and digital
Digital (109, 128, 43407)-net over F5, using
- net defined by OOA [i] based on linear OOA(5128, 43407, F5, 19, 19) (dual of [(43407, 19), 824605, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5128, 390664, F5, 19) (dual of [390664, 390536, 20]-code), using
- construction X applied to Ce(18) ⊂ 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(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(57, 39, F5, 4) (dual of [39, 32, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(57, 44, F5, 4) (dual of [44, 37, 5]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(5128, 390664, F5, 19) (dual of [390664, 390536, 20]-code), using
(109, 109+19, 298941)-Net over F5 — Digital
Digital (109, 128, 298941)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5128, 298941, F5, 19) (dual of [298941, 298813, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(5128, 390659, F5, 19) (dual of [390659, 390531, 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, 30, F5, 2) (dual of [30, 27, 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, 4, F5, 1) (dual of [4, 3, 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(5128, 390659, F5, 19) (dual of [390659, 390531, 20]-code), using
(109, 109+19, large)-Net in Base 5 — Upper bound on s
There is no (109, 128, large)-net in base 5, because
- 17 times m-reduction [i] would yield (109, 111, large)-net in base 5, but