Best Known (96, 96+13, s)-Nets in Base 7
(96, 96+13, 1921651)-Net over F7 — Constructive and digital
Digital (96, 109, 1921651)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (5, 11, 50)-net over F7, using
- base reduction for projective spaces (embedding PG(5,49) in PG(10,7)) for nets [i] based on digital (0, 6, 50)-net over F49, using
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 0 and N(F) ≥ 50, using
- the rational function field F49(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- base reduction for projective spaces (embedding PG(5,49) in PG(10,7)) for nets [i] based on digital (0, 6, 50)-net over F49, using
- digital (85, 98, 1921601)-net over F7, using
- net defined by OOA [i] based on linear OOA(798, 1921601, F7, 14, 13) (dual of [(1921601, 14), 26902316, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(798, 5764804, F7, 2, 13) (dual of [(5764804, 2), 11529510, 14]-NRT-code), using
- trace code [i] based on linear OOA(4949, 2882402, F49, 2, 13) (dual of [(2882402, 2), 5764755, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4949, 5764804, F49, 13) (dual of [5764804, 5764755, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4949, 5764805, F49, 13) (dual of [5764805, 5764756, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(4949, 5764801, F49, 13) (dual of [5764801, 5764752, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(4945, 5764801, F49, 12) (dual of [5764801, 5764756, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(490, 4, F49, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(4949, 5764805, F49, 13) (dual of [5764805, 5764756, 14]-code), using
- OOA 2-folding [i] based on linear OA(4949, 5764804, F49, 13) (dual of [5764804, 5764755, 14]-code), using
- trace code [i] based on linear OOA(4949, 2882402, F49, 2, 13) (dual of [(2882402, 2), 5764755, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(798, 5764804, F7, 2, 13) (dual of [(5764804, 2), 11529510, 14]-NRT-code), using
- net defined by OOA [i] based on linear OOA(798, 1921601, F7, 14, 13) (dual of [(1921601, 14), 26902316, 14]-NRT-code), using
- digital (5, 11, 50)-net over F7, using
(96, 96+13, large)-Net over F7 — Digital
Digital (96, 109, large)-net over F7, using
- t-expansion [i] based on digital (94, 109, large)-net over F7, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(7109, large, F7, 15) (dual of [large, large−109, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 20176804 | 718−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(7109, large, F7, 15) (dual of [large, large−109, 16]-code), using
(96, 96+13, large)-Net in Base 7 — Upper bound on s
There is no (96, 109, large)-net in base 7, because
- 11 times m-reduction [i] would yield (96, 98, large)-net in base 7, but