Best Known (92−18, 92, s)-Nets in Base 5
(92−18, 92, 1739)-Net over F5 — Constructive and digital
Digital (74, 92, 1739)-net over F5, using
- 51 times duplication [i] based on digital (73, 91, 1739)-net over F5, using
- net defined by OOA [i] based on linear OOA(591, 1739, F5, 18, 18) (dual of [(1739, 18), 31211, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(591, 15651, F5, 18) (dual of [15651, 15560, 19]-code), using
- construction XX applied to Ce(17) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- linear OA(585, 15625, F5, 18) (dual of [15625, 15540, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(561, 15625, F5, 13) (dual of [15625, 15564, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(54, 24, F5, 3) (dual of [24, 20, 4]-code or 24-cap in PG(3,5)), using
- linear OA(50, 2, F5, 0) (dual of [2, 2, 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 XX applied to Ce(17) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- OA 9-folding and stacking [i] based on linear OA(591, 15651, F5, 18) (dual of [15651, 15560, 19]-code), using
- net defined by OOA [i] based on linear OOA(591, 1739, F5, 18, 18) (dual of [(1739, 18), 31211, 19]-NRT-code), using
(92−18, 92, 15657)-Net over F5 — Digital
Digital (74, 92, 15657)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(592, 15657, F5, 18) (dual of [15657, 15565, 19]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(591, 15655, F5, 18) (dual of [15655, 15564, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- linear OA(585, 15625, F5, 18) (dual of [15625, 15540, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(561, 15625, F5, 13) (dual of [15625, 15564, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(56, 30, F5, 4) (dual of [30, 24, 5]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- linear OA(591, 15656, F5, 17) (dual of [15656, 15565, 18]-code), using Gilbert–Varšamov bound and bm = 591 > Vbs−1(k−1) = 2651 981039 445899 085248 112238 386894 825342 529785 192013 710595 221165 [i]
- linear OA(50, 1, F5, 0) (dual of [1, 1, 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(591, 15655, F5, 18) (dual of [15655, 15564, 19]-code), using
- construction X with Varšamov bound [i] based on
(92−18, 92, large)-Net in Base 5 — Upper bound on s
There is no (74, 92, large)-net in base 5, because
- 16 times m-reduction [i] would yield (74, 76, large)-net in base 5, but