Best Known (117−18, 117, s)-Nets in Base 5
(117−18, 117, 43405)-Net over F5 — Constructive and digital
Digital (99, 117, 43405)-net over F5, using
- 51 times duplication [i] based on digital (98, 116, 43405)-net over F5, using
- net defined by OOA [i] based on linear OOA(5116, 43405, F5, 18, 18) (dual of [(43405, 18), 781174, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(5116, 390645, F5, 18) (dual of [390645, 390529, 19]-code), using
- construction XX applied to Ce(17) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- linear OA(5113, 390625, F5, 18) (dual of [390625, 390512, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(51, 18, F5, 1) (dual of [18, 17, 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(51, 2, F5, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(17) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- OA 9-folding and stacking [i] based on linear OA(5116, 390645, F5, 18) (dual of [390645, 390529, 19]-code), using
- net defined by OOA [i] based on linear OOA(5116, 43405, F5, 18, 18) (dual of [(43405, 18), 781174, 19]-NRT-code), using
(117−18, 117, 198604)-Net over F5 — Digital
Digital (99, 117, 198604)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5117, 198604, F5, 18) (dual of [198604, 198487, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(5117, 390651, F5, 18) (dual of [390651, 390534, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(5113, 390625, F5, 18) (dual of [390625, 390512, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(54, 26, F5, 3) (dual of [26, 22, 4]-code or 26-cap in PG(3,5)), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(5117, 390651, F5, 18) (dual of [390651, 390534, 19]-code), using
(117−18, 117, large)-Net in Base 5 — Upper bound on s
There is no (99, 117, large)-net in base 5, because
- 16 times m-reduction [i] would yield (99, 101, large)-net in base 5, but