Best Known (115, 115+24, s)-Nets in Base 5
(115, 115+24, 6512)-Net over F5 — Constructive and digital
Digital (115, 139, 6512)-net over F5, using
- 52 times duplication [i] based on digital (113, 137, 6512)-net over F5, using
- net defined by OOA [i] based on linear OOA(5137, 6512, F5, 24, 24) (dual of [(6512, 24), 156151, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(5137, 78144, F5, 24) (dual of [78144, 78007, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(5137, 78149, F5, 24) (dual of [78149, 78012, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- linear OA(5134, 78125, F5, 24) (dual of [78125, 77991, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(5113, 78125, F5, 21) (dual of [78125, 78012, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 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]
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(5137, 78149, F5, 24) (dual of [78149, 78012, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(5137, 78144, F5, 24) (dual of [78144, 78007, 25]-code), using
- net defined by OOA [i] based on linear OOA(5137, 6512, F5, 24, 24) (dual of [(6512, 24), 156151, 25]-NRT-code), using
(115, 115+24, 54843)-Net over F5 — Digital
Digital (115, 139, 54843)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5139, 54843, F5, 24) (dual of [54843, 54704, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(5139, 78152, F5, 24) (dual of [78152, 78013, 25]-code), using
- construction XX applied to Ce(23) ⊂ Ce(20) ⊂ Ce(18) [i] based on
- linear OA(5134, 78125, F5, 24) (dual of [78125, 77991, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(5113, 78125, F5, 21) (dual of [78125, 78012, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(5106, 78125, F5, 19) (dual of [78125, 78019, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(53, 25, F5, 2) (dual of [25, 22, 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, 2, F5, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(23) ⊂ Ce(20) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(5139, 78152, F5, 24) (dual of [78152, 78013, 25]-code), using
(115, 115+24, large)-Net in Base 5 — Upper bound on s
There is no (115, 139, large)-net in base 5, because
- 22 times m-reduction [i] would yield (115, 117, large)-net in base 5, but