Best Known (122−26, 122, s)-Nets in Base 5
(122−26, 122, 1202)-Net over F5 — Constructive and digital
Digital (96, 122, 1202)-net over F5, using
- net defined by OOA [i] based on linear OOA(5122, 1202, F5, 26, 26) (dual of [(1202, 26), 31130, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(5122, 15626, F5, 26) (dual of [15626, 15504, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(5122, 15632, F5, 26) (dual of [15632, 15510, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(23) [i] based on
- linear OA(5121, 15625, F5, 26) (dual of [15625, 15504, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(5115, 15625, F5, 24) (dual of [15625, 15510, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(25) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(5122, 15632, F5, 26) (dual of [15632, 15510, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(5122, 15626, F5, 26) (dual of [15626, 15504, 27]-code), using
(122−26, 122, 8173)-Net over F5 — Digital
Digital (96, 122, 8173)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5122, 8173, F5, 26) (dual of [8173, 8051, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(5122, 15632, F5, 26) (dual of [15632, 15510, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(23) [i] based on
- linear OA(5121, 15625, F5, 26) (dual of [15625, 15504, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(5115, 15625, F5, 24) (dual of [15625, 15510, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(25) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(5122, 15632, F5, 26) (dual of [15632, 15510, 27]-code), using
(122−26, 122, 5139372)-Net in Base 5 — Upper bound on s
There is no (96, 122, 5139373)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 18 807953 143120 372812 330169 658872 894984 249510 290132 179710 897233 545084 889967 469377 848645 > 5122 [i]