Best Known (82, 101, s)-Nets in Base 7
(82, 101, 13075)-Net over F7 — Constructive and digital
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
(82, 101, 111923)-Net over F7 — Digital
Digital (82, 101, 111923)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7101, 111923, F7, 19) (dual of [111923, 111822, 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
(82, 101, large)-Net in Base 7 — Upper bound on s
There is no (82, 101, large)-net in base 7, because
- 17 times m-reduction [i] would yield (82, 84, large)-net in base 7, but