Best Known (97, 97+18, s)-Nets in Base 5
(97, 97+18, 43404)-Net over F5 — Constructive and digital
Digital (97, 115, 43404)-net over F5, using
- 51 times duplication [i] based on digital (96, 114, 43404)-net over F5, using
- net defined by OOA [i] based on linear OOA(5114, 43404, F5, 18, 18) (dual of [(43404, 18), 781158, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(5114, 390636, F5, 18) (dual of [390636, 390522, 19]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(5114, 390642, F5, 18) (dual of [390642, 390528, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(5114, 390636, F5, 18) (dual of [390636, 390522, 19]-code), using
- net defined by OOA [i] based on linear OOA(5114, 43404, F5, 18, 18) (dual of [(43404, 18), 781158, 19]-NRT-code), using
(97, 97+18, 195321)-Net over F5 — Digital
Digital (97, 115, 195321)-net over F5, using
- 51 times duplication [i] based on digital (96, 114, 195321)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [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
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5114, 195321, F5, 2, 18) (dual of [(195321, 2), 390528, 19]-NRT-code), using
(97, 97+18, large)-Net in Base 5 — Upper bound on s
There is no (97, 115, large)-net in base 5, because
- 16 times m-reduction [i] would yield (97, 99, large)-net in base 5, but