Best Known (58, 74, s)-Nets in Base 5
(58, 74, 1954)-Net over F5 — Constructive and digital
Digital (58, 74, 1954)-net over F5, using
- net defined by OOA [i] based on linear OOA(574, 1954, F5, 16, 16) (dual of [(1954, 16), 31190, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(574, 15632, F5, 16) (dual of [15632, 15558, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(573, 15625, F5, 16) (dual of [15625, 15552, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(567, 15625, F5, 14) (dual of [15625, 15558, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(51, 7, F5, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- OA 8-folding and stacking [i] based on linear OA(574, 15632, F5, 16) (dual of [15632, 15558, 17]-code), using
(58, 74, 7816)-Net over F5 — Digital
Digital (58, 74, 7816)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(574, 7816, F5, 2, 16) (dual of [(7816, 2), 15558, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(574, 15632, F5, 16) (dual of [15632, 15558, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(573, 15625, F5, 16) (dual of [15625, 15552, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(567, 15625, F5, 14) (dual of [15625, 15558, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(51, 7, F5, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- OOA 2-folding [i] based on linear OA(574, 15632, F5, 16) (dual of [15632, 15558, 17]-code), using
(58, 74, 2748537)-Net in Base 5 — Upper bound on s
There is no (58, 74, 2748538)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 5293 955972 518118 910229 770108 031333 059939 981997 448065 > 574 [i]