Best Known (98, 98+18, s)-Nets in Base 5
(98, 98+18, 43405)-Net over F5 — Constructive and digital
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
(98, 98+18, 195322)-Net over F5 — Digital
Digital (98, 116, 195322)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5116, 195322, F5, 2, 18) (dual of [(195322, 2), 390528, 19]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(5114, 195321, F5, 2, 18) (dual of [(195321, 2), 390528, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5114, 390642, F5, 18) (dual of [390642, 390528, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [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(51, 17, F5, 1) (dual of [17, 16, 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(17) ⊂ Ce(15) [i] based on
- OOA 2-folding [i] based on linear OA(5114, 390642, F5, 18) (dual of [390642, 390528, 19]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(5114, 195321, F5, 2, 18) (dual of [(195321, 2), 390528, 19]-NRT-code), using
(98, 98+18, large)-Net in Base 5 — Upper bound on s
There is no (98, 116, large)-net in base 5, because
- 16 times m-reduction [i] would yield (98, 100, large)-net in base 5, but