Best Known (88, 88+13, s)-Nets in Base 7
(88, 88+13, 1921603)-Net over F7 — Constructive and digital
Digital (88, 101, 1921603)-net over F7, using
- 71 times duplication [i] based on digital (87, 100, 1921603)-net over F7, using
- net defined by OOA [i] based on linear OOA(7100, 1921603, F7, 14, 13) (dual of [(1921603, 14), 26902342, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(7100, 5764810, F7, 2, 13) (dual of [(5764810, 2), 11529520, 14]-NRT-code), using
- trace code [i] based on linear OOA(4950, 2882405, F49, 2, 13) (dual of [(2882405, 2), 5764760, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4950, 5764810, F49, 13) (dual of [5764810, 5764760, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4950, 5764811, F49, 13) (dual of [5764811, 5764761, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(4949, 5764802, F49, 13) (dual of [5764802, 5764753, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 5764802 | 498−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(4941, 5764802, F49, 11) (dual of [5764802, 5764761, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 5764802 | 498−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(491, 9, F49, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(491, s, F49, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4950, 5764811, F49, 13) (dual of [5764811, 5764761, 14]-code), using
- OOA 2-folding [i] based on linear OA(4950, 5764810, F49, 13) (dual of [5764810, 5764760, 14]-code), using
- trace code [i] based on linear OOA(4950, 2882405, F49, 2, 13) (dual of [(2882405, 2), 5764760, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(7100, 5764810, F7, 2, 13) (dual of [(5764810, 2), 11529520, 14]-NRT-code), using
- net defined by OOA [i] based on linear OOA(7100, 1921603, F7, 14, 13) (dual of [(1921603, 14), 26902342, 14]-NRT-code), using
(88, 88+13, large)-Net over F7 — Digital
Digital (88, 101, large)-net over F7, using
- 73 times duplication [i] based on digital (85, 98, large)-net over F7, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(798, large, F7, 13) (dual of [large, large−98, 14]-code), using
- trace code [i] based on linear OA(4949, 5764802, F49, 13) (dual of [5764802, 5764753, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 5764802 | 498−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- trace code [i] based on linear OA(4949, 5764802, F49, 13) (dual of [5764802, 5764753, 14]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(798, large, F7, 13) (dual of [large, large−98, 14]-code), using
(88, 88+13, large)-Net in Base 7 — Upper bound on s
There is no (88, 101, large)-net in base 7, because
- 11 times m-reduction [i] would yield (88, 90, large)-net in base 7, but