Best Known (83−14, 83, s)-Nets in Base 5
(83−14, 83, 11164)-Net over F5 — Constructive and digital
Digital (69, 83, 11164)-net over F5, using
- 52 times duplication [i] based on digital (67, 81, 11164)-net over F5, using
- net defined by OOA [i] based on linear OOA(581, 11164, F5, 14, 14) (dual of [(11164, 14), 156215, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(581, 78148, F5, 14) (dual of [78148, 78067, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(581, 78149, F5, 14) (dual of [78149, 78068, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- linear OA(578, 78125, F5, 14) (dual of [78125, 78047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(557, 78125, F5, 11) (dual of [78125, 78068, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(13) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(581, 78149, F5, 14) (dual of [78149, 78068, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(581, 78148, F5, 14) (dual of [78148, 78067, 15]-code), using
- net defined by OOA [i] based on linear OOA(581, 11164, F5, 14, 14) (dual of [(11164, 14), 156215, 15]-NRT-code), using
(83−14, 83, 78152)-Net over F5 — Digital
Digital (69, 83, 78152)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(583, 78152, F5, 14) (dual of [78152, 78069, 15]-code), using
- construction XX applied to Ce(13) ⊂ Ce(10) ⊂ Ce(8) [i] based on
- linear OA(578, 78125, F5, 14) (dual of [78125, 78047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(557, 78125, F5, 11) (dual of [78125, 78068, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(550, 78125, F5, 9) (dual of [78125, 78075, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(13) ⊂ Ce(10) ⊂ Ce(8) [i] based on
(83−14, 83, large)-Net in Base 5 — Upper bound on s
There is no (69, 83, large)-net in base 5, because
- 12 times m-reduction [i] would yield (69, 71, large)-net in base 5, but