Best Known (140−22, 140, s)-Nets in Base 5
(140−22, 140, 35513)-Net over F5 — Constructive and digital
Digital (118, 140, 35513)-net over F5, using
- net defined by OOA [i] based on linear OOA(5140, 35513, F5, 22, 22) (dual of [(35513, 22), 781146, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(5140, 390643, F5, 22) (dual of [390643, 390503, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(5140, 390644, F5, 22) (dual of [390644, 390504, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- linear OA(5137, 390625, F5, 22) (dual of [390625, 390488, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(5121, 390625, F5, 19) (dual of [390625, 390504, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(21) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(5140, 390644, F5, 22) (dual of [390644, 390504, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(5140, 390643, F5, 22) (dual of [390643, 390503, 23]-code), using
(140−22, 140, 195322)-Net over F5 — Digital
Digital (118, 140, 195322)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5140, 195322, F5, 2, 22) (dual of [(195322, 2), 390504, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5140, 390644, F5, 22) (dual of [390644, 390504, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- linear OA(5137, 390625, F5, 22) (dual of [390625, 390488, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(5121, 390625, F5, 19) (dual of [390625, 390504, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(21) ⊂ Ce(18) [i] based on
- OOA 2-folding [i] based on linear OA(5140, 390644, F5, 22) (dual of [390644, 390504, 23]-code), using
(140−22, 140, large)-Net in Base 5 — Upper bound on s
There is no (118, 140, large)-net in base 5, because
- 20 times m-reduction [i] would yield (118, 120, large)-net in base 5, but