Best Known (40, 51, s)-Nets in Base 7
(40, 51, 3370)-Net over F7 — Constructive and digital
Digital (40, 51, 3370)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 8)-net over F7, using
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 0 and N(F) ≥ 8, using
- the rational function field F7(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- digital (35, 46, 3362)-net over F7, using
- net defined by OOA [i] based on linear OOA(746, 3362, F7, 11, 11) (dual of [(3362, 11), 36936, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(746, 16811, F7, 11) (dual of [16811, 16765, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(746, 16812, F7, 11) (dual of [16812, 16766, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- 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(70, 5, F7, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(746, 16812, F7, 11) (dual of [16812, 16766, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(746, 16811, F7, 11) (dual of [16811, 16765, 12]-code), using
- net defined by OOA [i] based on linear OOA(746, 3362, F7, 11, 11) (dual of [(3362, 11), 36936, 12]-NRT-code), using
- digital (0, 5, 8)-net over F7, using
(40, 51, 16829)-Net over F7 — Digital
Digital (40, 51, 16829)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(751, 16829, F7, 11) (dual of [16829, 16778, 12]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(749, 16825, F7, 11) (dual of [16825, 16776, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- 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(731, 16807, F7, 8) (dual of [16807, 16776, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(73, 18, F7, 2) (dual of [18, 15, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(749, 16827, F7, 10) (dual of [16827, 16778, 11]-code), using Gilbert–Varšamov bound and bm = 749 > Vbs−1(k−1) = 2995 956949 232432 337061 320965 202951 521137 [i]
- linear OA(70, 2, F7, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(749, 16825, F7, 11) (dual of [16825, 16776, 12]-code), using
- construction X with Varšamov bound [i] based on
(40, 51, large)-Net in Base 7 — Upper bound on s
There is no (40, 51, large)-net in base 7, because
- 9 times m-reduction [i] would yield (40, 42, large)-net in base 7, but