Best Known (66−12, 66, s)-Nets in Base 5
(66−12, 66, 13022)-Net over F5 — Constructive and digital
Digital (54, 66, 13022)-net over F5, using
- 52 times duplication [i] based on digital (52, 64, 13022)-net over F5, using
- net defined by OOA [i] based on linear OOA(564, 13022, F5, 12, 12) (dual of [(13022, 12), 156200, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(564, 78132, F5, 12) (dual of [78132, 78068, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(564, 78125, F5, 12) (dual of [78125, 78061, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(50, 7, F5, 0) (dual of [7, 7, 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 X applied to Ce(11) ⊂ Ce(10) [i] based on
- OA 6-folding and stacking [i] based on linear OA(564, 78132, F5, 12) (dual of [78132, 78068, 13]-code), using
- net defined by OOA [i] based on linear OOA(564, 13022, F5, 12, 12) (dual of [(13022, 12), 156200, 13]-NRT-code), using
(66−12, 66, 39550)-Net over F5 — Digital
Digital (54, 66, 39550)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(566, 39550, F5, 12) (dual of [39550, 39484, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(566, 78135, F5, 12) (dual of [78135, 78069, 13]-code), using
- construction XX applied to Ce(11) ⊂ Ce(10) ⊂ Ce(8) [i] based on
- linear OA(564, 78125, F5, 12) (dual of [78125, 78061, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(50, 8, F5, 0) (dual of [8, 8, 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
- 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(11) ⊂ Ce(10) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(566, 78135, F5, 12) (dual of [78135, 78069, 13]-code), using
(66−12, 66, large)-Net in Base 5 — Upper bound on s
There is no (54, 66, large)-net in base 5, because
- 10 times m-reduction [i] would yield (54, 56, large)-net in base 5, but