Best Known (90, 90+14, s)-Nets in Base 5
(90, 90+14, 279022)-Net over F5 — Constructive and digital
Digital (90, 104, 279022)-net over F5, using
- 51 times duplication [i] based on digital (89, 103, 279022)-net over F5, using
- net defined by OOA [i] based on linear OOA(5103, 279022, F5, 14, 14) (dual of [(279022, 14), 3906205, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(5103, 1953154, F5, 14) (dual of [1953154, 1953051, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(5103, 1953155, F5, 14) (dual of [1953155, 1953052, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- linear OA(5100, 1953125, F5, 14) (dual of [1953125, 1953025, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(573, 1953125, F5, 11) (dual of [1953125, 1953052, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(53, 30, F5, 2) (dual of [30, 27, 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
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(5103, 1953155, F5, 14) (dual of [1953155, 1953052, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(5103, 1953154, F5, 14) (dual of [1953154, 1953051, 15]-code), using
- net defined by OOA [i] based on linear OOA(5103, 279022, F5, 14, 14) (dual of [(279022, 14), 3906205, 15]-NRT-code), using
(90, 90+14, 1320661)-Net over F5 — Digital
Digital (90, 104, 1320661)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5104, 1320661, F5, 14) (dual of [1320661, 1320557, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(5104, 1953156, F5, 14) (dual of [1953156, 1953052, 15]-code), using
- 1 times code embedding in larger space [i] based on linear OA(5103, 1953155, F5, 14) (dual of [1953155, 1953052, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- linear OA(5100, 1953125, F5, 14) (dual of [1953125, 1953025, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(573, 1953125, F5, 11) (dual of [1953125, 1953052, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(53, 30, F5, 2) (dual of [30, 27, 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
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(5103, 1953155, F5, 14) (dual of [1953155, 1953052, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(5104, 1953156, F5, 14) (dual of [1953156, 1953052, 15]-code), using
(90, 90+14, large)-Net in Base 5 — Upper bound on s
There is no (90, 104, large)-net in base 5, because
- 12 times m-reduction [i] would yield (90, 92, large)-net in base 5, but