Best Known (84, 84+19, s)-Nets in Base 7
(84, 84+19, 13075)-Net over F7 — Constructive and digital
Digital (84, 103, 13075)-net over F7, using
- 72 times duplication [i] based on digital (82, 101, 13075)-net over F7, using
- net defined by OOA [i] based on linear OOA(7101, 13075, F7, 19, 19) (dual of [(13075, 19), 248324, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(7101, 117676, F7, 19) (dual of [117676, 117575, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(7101, 117678, F7, 19) (dual of [117678, 117577, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- linear OA(797, 117650, F7, 19) (dual of [117650, 117553, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 712−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (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(74, 28, F7, 3) (dual of [28, 24, 4]-code or 28-cap in PG(3,7)), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(7101, 117678, F7, 19) (dual of [117678, 117577, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(7101, 117676, F7, 19) (dual of [117676, 117575, 20]-code), using
- net defined by OOA [i] based on linear OOA(7101, 13075, F7, 19, 19) (dual of [(13075, 19), 248324, 20]-NRT-code), using
(84, 84+19, 117682)-Net over F7 — Digital
Digital (84, 103, 117682)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7103, 117682, F7, 19) (dual of [117682, 117579, 20]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(7101, 117678, F7, 19) (dual of [117678, 117577, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- linear OA(797, 117650, F7, 19) (dual of [117650, 117553, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 712−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (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(74, 28, F7, 3) (dual of [28, 24, 4]-code or 28-cap in PG(3,7)), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- linear OA(7101, 117680, F7, 18) (dual of [117680, 117579, 19]-code), using Gilbert–Varšamov bound and bm = 7101 > Vbs−1(k−1) = 7 566477 494903 534340 314654 930375 305747 835375 897299 411454 002532 354326 885301 636888 100151 [i]
- linear OA(70, 2, F7, 0) (dual of [2, 2, 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(7101, 117678, F7, 19) (dual of [117678, 117577, 20]-code), using
- construction X with Varšamov bound [i] based on
(84, 84+19, large)-Net in Base 7 — Upper bound on s
There is no (84, 103, large)-net in base 7, because
- 17 times m-reduction [i] would yield (84, 86, large)-net in base 7, but