Best Known (72−17, 72, s)-Nets in Base 7
(72−17, 72, 2102)-Net over F7 — Constructive and digital
Digital (55, 72, 2102)-net over F7, using
- net defined by OOA [i] based on linear OOA(772, 2102, F7, 17, 17) (dual of [(2102, 17), 35662, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(772, 16817, F7, 17) (dual of [16817, 16745, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(772, 16819, F7, 17) (dual of [16819, 16747, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(771, 16808, F7, 17) (dual of [16808, 16737, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 16808 | 710−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(761, 16808, F7, 15) (dual of [16808, 16747, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 16808 | 710−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(71, 11, F7, 1) (dual of [11, 10, 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
- discarding factors / shortening the dual code based on linear OA(772, 16819, F7, 17) (dual of [16819, 16747, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(772, 16817, F7, 17) (dual of [16817, 16745, 18]-code), using
(72−17, 72, 10700)-Net over F7 — Digital
Digital (55, 72, 10700)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(772, 10700, F7, 17) (dual of [10700, 10628, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(772, 16819, F7, 17) (dual of [16819, 16747, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(771, 16808, F7, 17) (dual of [16808, 16737, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 16808 | 710−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(761, 16808, F7, 15) (dual of [16808, 16747, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 16808 | 710−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(71, 11, F7, 1) (dual of [11, 10, 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
- discarding factors / shortening the dual code based on linear OA(772, 16819, F7, 17) (dual of [16819, 16747, 18]-code), using
(72−17, 72, large)-Net in Base 7 — Upper bound on s
There is no (55, 72, large)-net in base 7, because
- 15 times m-reduction [i] would yield (55, 57, large)-net in base 7, but