Best Known (44, 44+12, s)-Nets in Base 7
(44, 44+12, 2805)-Net over F7 — Constructive and digital
Digital (44, 56, 2805)-net over F7, using
- 71 times duplication [i] based on digital (43, 55, 2805)-net over F7, using
- net defined by OOA [i] based on linear OOA(755, 2805, F7, 12, 12) (dual of [(2805, 12), 33605, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(755, 16830, F7, 12) (dual of [16830, 16775, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(755, 16831, F7, 12) (dual of [16831, 16776, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(751, 16807, F7, 12) (dual of [16807, 16756, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(731, 16807, F7, 8) (dual of [16807, 16776, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(74, 24, F7, 3) (dual of [24, 20, 4]-code or 24-cap in PG(3,7)), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(755, 16831, F7, 12) (dual of [16831, 16776, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(755, 16830, F7, 12) (dual of [16830, 16775, 13]-code), using
- net defined by OOA [i] based on linear OOA(755, 2805, F7, 12, 12) (dual of [(2805, 12), 33605, 13]-NRT-code), using
(44, 44+12, 16833)-Net over F7 — Digital
Digital (44, 56, 16833)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(756, 16833, F7, 12) (dual of [16833, 16777, 13]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(755, 16831, F7, 12) (dual of [16831, 16776, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(751, 16807, F7, 12) (dual of [16807, 16756, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(731, 16807, F7, 8) (dual of [16807, 16776, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(74, 24, F7, 3) (dual of [24, 20, 4]-code or 24-cap in PG(3,7)), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(755, 16832, F7, 11) (dual of [16832, 16777, 12]-code), using Gilbert–Varšamov bound and bm = 755 > Vbs−1(k−1) = 30 320079 540736 878444 288575 074557 682732 239799 [i]
- linear OA(70, 1, F7, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(755, 16831, F7, 12) (dual of [16831, 16776, 13]-code), using
- construction X with Varšamov bound [i] based on
(44, 44+12, large)-Net in Base 7 — Upper bound on s
There is no (44, 56, large)-net in base 7, because
- 10 times m-reduction [i] would yield (44, 46, large)-net in base 7, but