Best Known (82, 106, s)-Nets in Base 7
(82, 106, 1402)-Net over F7 — Constructive and digital
Digital (82, 106, 1402)-net over F7, using
- 71 times duplication [i] based on digital (81, 105, 1402)-net over F7, using
- net defined by OOA [i] based on linear OOA(7105, 1402, F7, 24, 24) (dual of [(1402, 24), 33543, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(7105, 16824, F7, 24) (dual of [16824, 16719, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(7105, 16826, F7, 24) (dual of [16826, 16721, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(19) [i] based on
- linear OA(7101, 16807, F7, 24) (dual of [16807, 16706, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(786, 16807, F7, 20) (dual of [16807, 16721, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(74, 19, F7, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,7)), using
- construction X applied to Ce(23) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(7105, 16826, F7, 24) (dual of [16826, 16721, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(7105, 16824, F7, 24) (dual of [16824, 16719, 25]-code), using
- net defined by OOA [i] based on linear OOA(7105, 1402, F7, 24, 24) (dual of [(1402, 24), 33543, 25]-NRT-code), using
(82, 106, 16284)-Net over F7 — Digital
Digital (82, 106, 16284)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7106, 16284, F7, 24) (dual of [16284, 16178, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(7106, 16828, F7, 24) (dual of [16828, 16722, 25]-code), using
- construction XX applied to Ce(23) ⊂ Ce(19) ⊂ Ce(18) [i] based on
- linear OA(7101, 16807, F7, 24) (dual of [16807, 16706, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(786, 16807, F7, 20) (dual of [16807, 16721, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(781, 16807, F7, 19) (dual of [16807, 16726, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(74, 20, F7, 3) (dual of [20, 16, 4]-code or 20-cap in PG(3,7)), using
- linear OA(70, 1, F7, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(23) ⊂ Ce(19) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(7106, 16828, F7, 24) (dual of [16828, 16722, 25]-code), using
(82, 106, large)-Net in Base 7 — Upper bound on s
There is no (82, 106, large)-net in base 7, because
- 22 times m-reduction [i] would yield (82, 84, large)-net in base 7, but