Best Known (72, 72+20, s)-Nets in Base 7
(72, 72+20, 1683)-Net over F7 — Constructive and digital
Digital (72, 92, 1683)-net over F7, using
- 72 times duplication [i] based on digital (70, 90, 1683)-net over F7, using
- net defined by OOA [i] based on linear OOA(790, 1683, F7, 20, 20) (dual of [(1683, 20), 33570, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(790, 16830, F7, 20) (dual of [16830, 16740, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(790, 16831, F7, 20) (dual of [16831, 16741, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- 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(766, 16807, F7, 16) (dual of [16807, 16741, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(19) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(790, 16831, F7, 20) (dual of [16831, 16741, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(790, 16830, F7, 20) (dual of [16830, 16740, 21]-code), using
- net defined by OOA [i] based on linear OOA(790, 1683, F7, 20, 20) (dual of [(1683, 20), 33570, 21]-NRT-code), using
(72, 72+20, 16838)-Net over F7 — Digital
Digital (72, 92, 16838)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(792, 16838, F7, 20) (dual of [16838, 16746, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- 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(761, 16807, F7, 15) (dual of [16807, 16746, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(76, 31, F7, 4) (dual of [31, 25, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(76, 42, F7, 4) (dual of [42, 36, 5]-code), using
- 1 times truncation [i] based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(76, 42, F7, 4) (dual of [42, 36, 5]-code), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
(72, 72+20, large)-Net in Base 7 — Upper bound on s
There is no (72, 92, large)-net in base 7, because
- 18 times m-reduction [i] would yield (72, 74, large)-net in base 7, but