Best Known (122, 122+22, s)-Nets in Base 5
(122, 122+22, 35514)-Net over F5 — Constructive and digital
Digital (122, 144, 35514)-net over F5, using
- 52 times duplication [i] based on digital (120, 142, 35514)-net over F5, using
- net defined by OOA [i] based on linear OOA(5142, 35514, F5, 22, 22) (dual of [(35514, 22), 781166, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(5142, 390654, F5, 22) (dual of [390654, 390512, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(17) [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(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(55, 29, F5, 3) (dual of [29, 24, 4]-code or 29-cap in PG(4,5)), using
- discarding factors / shortening the dual code based on linear OA(55, 42, F5, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,5)), using
- construction X applied to Ce(21) ⊂ Ce(17) [i] based on
- OA 11-folding and stacking [i] based on linear OA(5142, 390654, F5, 22) (dual of [390654, 390512, 23]-code), using
- net defined by OOA [i] based on linear OOA(5142, 35514, F5, 22, 22) (dual of [(35514, 22), 781166, 23]-NRT-code), using
(122, 122+22, 206468)-Net over F5 — Digital
Digital (122, 144, 206468)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5144, 206468, F5, 22) (dual of [206468, 206324, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(5144, 390658, F5, 22) (dual of [390658, 390514, 23]-code), using
- construction XX applied to Ce(21) ⊂ Ce(17) ⊂ Ce(16) [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(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(5105, 390625, F5, 17) (dual of [390625, 390520, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(55, 31, F5, 3) (dual of [31, 26, 4]-code or 31-cap in PG(4,5)), using
- discarding factors / shortening the dual code based on linear OA(55, 42, F5, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,5)), using
- linear OA(50, 2, F5, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction XX applied to Ce(21) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(5144, 390658, F5, 22) (dual of [390658, 390514, 23]-code), using
(122, 122+22, large)-Net in Base 5 — Upper bound on s
There is no (122, 144, large)-net in base 5, because
- 20 times m-reduction [i] would yield (122, 124, large)-net in base 5, but