Best Known (58, 71, s)-Nets in Base 5
(58, 71, 13021)-Net over F5 — Constructive and digital
Digital (58, 71, 13021)-net over F5, using
- net defined by OOA [i] based on linear OOA(571, 13021, F5, 13, 13) (dual of [(13021, 13), 169202, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(571, 78127, F5, 13) (dual of [78127, 78056, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(571, 78132, F5, 13) (dual of [78132, 78061, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(571, 78125, F5, 13) (dual of [78125, 78054, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(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(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(571, 78132, F5, 13) (dual of [78132, 78061, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(571, 78127, F5, 13) (dual of [78127, 78056, 14]-code), using
(58, 71, 39066)-Net over F5 — Digital
Digital (58, 71, 39066)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(571, 39066, F5, 2, 13) (dual of [(39066, 2), 78061, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(571, 78132, F5, 13) (dual of [78132, 78061, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(571, 78125, F5, 13) (dual of [78125, 78054, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(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(12) ⊂ Ce(11) [i] based on
- OOA 2-folding [i] based on linear OA(571, 78132, F5, 13) (dual of [78132, 78061, 14]-code), using
(58, 71, large)-Net in Base 5 — Upper bound on s
There is no (58, 71, large)-net in base 5, because
- 11 times m-reduction [i] would yield (58, 60, large)-net in base 5, but