Best Known (115−21, 115, s)-Nets in Base 5
(115−21, 115, 7813)-Net over F5 — Constructive and digital
Digital (94, 115, 7813)-net over F5, using
- 51 times duplication [i] based on digital (93, 114, 7813)-net over F5, using
- net defined by OOA [i] based on linear OOA(5114, 7813, F5, 21, 21) (dual of [(7813, 21), 163959, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(5114, 78131, F5, 21) (dual of [78131, 78017, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(5114, 78133, F5, 21) (dual of [78133, 78019, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- 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(51, 8, F5, 1) (dual of [8, 7, 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(20) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(5114, 78133, F5, 21) (dual of [78133, 78019, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(5114, 78131, F5, 21) (dual of [78131, 78017, 22]-code), using
- net defined by OOA [i] based on linear OOA(5114, 7813, F5, 21, 21) (dual of [(7813, 21), 163959, 22]-NRT-code), using
(115−21, 115, 39067)-Net over F5 — Digital
Digital (94, 115, 39067)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5115, 39067, F5, 2, 21) (dual of [(39067, 2), 78019, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5115, 78134, F5, 21) (dual of [78134, 78019, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(5115, 78135, F5, 21) (dual of [78135, 78020, 22]-code), using
- construction XX applied to Ce(20) ⊂ Ce(18) ⊂ Ce(17) [i] based on
- 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(599, 78125, F5, 18) (dual of [78125, 78026, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(51, 9, F5, 1) (dual of [9, 8, 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(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(20) ⊂ Ce(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(5115, 78135, F5, 21) (dual of [78135, 78020, 22]-code), using
- OOA 2-folding [i] based on linear OA(5115, 78134, F5, 21) (dual of [78134, 78019, 22]-code), using
(115−21, 115, large)-Net in Base 5 — Upper bound on s
There is no (94, 115, large)-net in base 5, because
- 19 times m-reduction [i] would yield (94, 96, large)-net in base 5, but