Best Known (123−26, 123, s)-Nets in Base 5
(123−26, 123, 1202)-Net over F5 — Constructive and digital
Digital (97, 123, 1202)-net over F5, using
- 51 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(5122, 1202, F5, 26, 26) (dual of [(1202, 26), 31130, 27]-NRT-code), using
(123−26, 123, 8741)-Net over F5 — Digital
Digital (97, 123, 8741)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5123, 8741, F5, 26) (dual of [8741, 8618, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(5123, 15634, F5, 26) (dual of [15634, 15511, 27]-code), using
- construction XX applied to Ce(25) ⊂ Ce(23) ⊂ Ce(22) [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(5109, 15625, F5, 23) (dual of [15625, 15516, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(51, 8, F5, 1) (dual of [8, 7, 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
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(25) ⊂ Ce(23) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(5123, 15634, F5, 26) (dual of [15634, 15511, 27]-code), using
(123−26, 123, 5816705)-Net in Base 5 — Upper bound on s
There is no (97, 123, 5816706)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 94 039654 002991 724044 478810 862256 448657 221441 652260 588009 026478 870873 249083 042236 514825 > 5123 [i]