Best Known (89, 89+21, s)-Nets in Base 7
(89, 89+21, 11766)-Net over F7 — Constructive and digital
Digital (89, 110, 11766)-net over F7, using
- net defined by OOA [i] based on linear OOA(7110, 11766, F7, 21, 21) (dual of [(11766, 21), 246976, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(7110, 117661, F7, 21) (dual of [117661, 117551, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(7110, 117663, F7, 21) (dual of [117663, 117553, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(7109, 117650, F7, 21) (dual of [117650, 117541, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 712−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- 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(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([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(7110, 117663, F7, 21) (dual of [117663, 117553, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(7110, 117661, F7, 21) (dual of [117661, 117551, 22]-code), using
(89, 89+21, 93155)-Net over F7 — Digital
Digital (89, 110, 93155)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7110, 93155, F7, 21) (dual of [93155, 93045, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(7110, 117663, F7, 21) (dual of [117663, 117553, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(7109, 117650, F7, 21) (dual of [117650, 117541, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 712−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- 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(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([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(7110, 117663, F7, 21) (dual of [117663, 117553, 22]-code), using
(89, 89+21, large)-Net in Base 7 — Upper bound on s
There is no (89, 110, large)-net in base 7, because
- 19 times m-reduction [i] would yield (89, 91, large)-net in base 7, but