Best Known (51, 51+15, s)-Nets in Base 7
(51, 51+15, 2403)-Net over F7 — Constructive and digital
Digital (51, 66, 2403)-net over F7, using
- 71 times duplication [i] based on digital (50, 65, 2403)-net over F7, using
- net defined by OOA [i] based on linear OOA(765, 2403, F7, 15, 15) (dual of [(2403, 15), 35980, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(765, 16822, F7, 15) (dual of [16822, 16757, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- 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(751, 16808, F7, 11) (dual of [16808, 16757, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 16808 | 710−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(74, 14, F7, 3) (dual of [14, 10, 4]-code or 14-cap in PG(3,7)), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(765, 16822, F7, 15) (dual of [16822, 16757, 16]-code), using
- net defined by OOA [i] based on linear OOA(765, 2403, F7, 15, 15) (dual of [(2403, 15), 35980, 16]-NRT-code), using
(51, 51+15, 15868)-Net over F7 — Digital
Digital (51, 66, 15868)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(766, 15868, F7, 15) (dual of [15868, 15802, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(766, 16828, F7, 15) (dual of [16828, 16762, 16]-code), using
- construction XX applied to Ce(14) ⊂ Ce(10) ⊂ Ce(9) [i] based on
- linear OA(761, 16807, F7, 15) (dual of [16807, 16746, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(746, 16807, F7, 11) (dual of [16807, 16761, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(741, 16807, F7, 10) (dual of [16807, 16766, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(74, 20, F7, 3) (dual of [20, 16, 4]-code or 20-cap in PG(3,7)), using
- linear OA(70, 1, F7, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(14) ⊂ Ce(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(766, 16828, F7, 15) (dual of [16828, 16762, 16]-code), using
(51, 51+15, large)-Net in Base 7 — Upper bound on s
There is no (51, 66, large)-net in base 7, because
- 13 times m-reduction [i] would yield (51, 53, large)-net in base 7, but