Best Known (72, 89, s)-Nets in Base 7
(72, 89, 14708)-Net over F7 — Constructive and digital
Digital (72, 89, 14708)-net over F7, using
- 71 times duplication [i] based on digital (71, 88, 14708)-net over F7, using
- net defined by OOA [i] based on linear OOA(788, 14708, F7, 17, 17) (dual of [(14708, 17), 249948, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(788, 117665, F7, 17) (dual of [117665, 117577, 18]-code), using
- 2 times code embedding in larger space [i] based on linear OA(786, 117663, F7, 17) (dual of [117663, 117577, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(785, 117650, F7, 17) (dual of [117650, 117565, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 712−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(773, 117650, F7, 15) (dual of [117650, 117577, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 712−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(71, 13, F7, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(786, 117663, F7, 17) (dual of [117663, 117577, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(788, 117665, F7, 17) (dual of [117665, 117577, 18]-code), using
- net defined by OOA [i] based on linear OOA(788, 14708, F7, 17, 17) (dual of [(14708, 17), 249948, 18]-NRT-code), using
(72, 89, 97159)-Net over F7 — Digital
Digital (72, 89, 97159)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(789, 97159, F7, 17) (dual of [97159, 97070, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(789, 117671, F7, 17) (dual of [117671, 117582, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- linear OA(785, 117649, F7, 17) (dual of [117649, 117564, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(767, 117649, F7, 13) (dual of [117649, 117582, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(74, 22, F7, 3) (dual of [22, 18, 4]-code or 22-cap in PG(3,7)), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(789, 117671, F7, 17) (dual of [117671, 117582, 18]-code), using
(72, 89, large)-Net in Base 7 — Upper bound on s
There is no (72, 89, large)-net in base 7, because
- 15 times m-reduction [i] would yield (72, 74, large)-net in base 7, but