Best Known (99−15, 99, s)-Nets in Base 7
(99−15, 99, 823544)-Net over F7 — Constructive and digital
Digital (84, 99, 823544)-net over F7, using
- 71 times duplication [i] based on digital (83, 98, 823544)-net over F7, using
- net defined by OOA [i] based on linear OOA(798, 823544, F7, 15, 15) (dual of [(823544, 15), 12353062, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(798, 5764809, F7, 15) (dual of [5764809, 5764711, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(798, 5764810, F7, 15) (dual of [5764810, 5764712, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(797, 5764801, F7, 15) (dual of [5764801, 5764704, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(789, 5764801, F7, 13) (dual of [5764801, 5764712, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(71, 9, F7, 1) (dual of [9, 8, 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 Ce(14) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(798, 5764810, F7, 15) (dual of [5764810, 5764712, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(798, 5764809, F7, 15) (dual of [5764809, 5764711, 16]-code), using
- net defined by OOA [i] based on linear OOA(798, 823544, F7, 15, 15) (dual of [(823544, 15), 12353062, 16]-NRT-code), using
(99−15, 99, 2882406)-Net over F7 — Digital
Digital (84, 99, 2882406)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(799, 2882406, F7, 2, 15) (dual of [(2882406, 2), 5764713, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(799, 5764812, F7, 15) (dual of [5764812, 5764713, 16]-code), using
- construction XX applied to Ce(14) ⊂ Ce(12) ⊂ Ce(11) [i] based on
- linear OA(797, 5764801, F7, 15) (dual of [5764801, 5764704, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(789, 5764801, F7, 13) (dual of [5764801, 5764712, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(781, 5764801, F7, 12) (dual of [5764801, 5764720, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(71, 10, F7, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), 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(12) ⊂ Ce(11) [i] based on
- OOA 2-folding [i] based on linear OA(799, 5764812, F7, 15) (dual of [5764812, 5764713, 16]-code), using
(99−15, 99, large)-Net in Base 7 — Upper bound on s
There is no (84, 99, large)-net in base 7, because
- 13 times m-reduction [i] would yield (84, 86, large)-net in base 7, but