Best Known (84, 84+16, s)-Nets in Base 5
(84, 84+16, 48830)-Net over F5 — Constructive and digital
Digital (84, 100, 48830)-net over F5, using
- net defined by OOA [i] based on linear OOA(5100, 48830, F5, 16, 16) (dual of [(48830, 16), 781180, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(5100, 390640, F5, 16) (dual of [390640, 390540, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(5100, 390644, F5, 16) (dual of [390644, 390544, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- 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(581, 390625, F5, 13) (dual of [390625, 390544, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(53, 19, F5, 2) (dual of [19, 16, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 24 = 52−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(5100, 390644, F5, 16) (dual of [390644, 390544, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(5100, 390640, F5, 16) (dual of [390640, 390540, 17]-code), using
(84, 84+16, 195322)-Net over F5 — Digital
Digital (84, 100, 195322)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5100, 195322, F5, 2, 16) (dual of [(195322, 2), 390544, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5100, 390644, F5, 16) (dual of [390644, 390544, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- 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(581, 390625, F5, 13) (dual of [390625, 390544, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(53, 19, F5, 2) (dual of [19, 16, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 24 = 52−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- OOA 2-folding [i] based on linear OA(5100, 390644, F5, 16) (dual of [390644, 390544, 17]-code), using
(84, 84+16, large)-Net in Base 5 — Upper bound on s
There is no (84, 100, large)-net in base 5, because
- 14 times m-reduction [i] would yield (84, 86, large)-net in base 5, but