Best Known (49, 63, s)-Nets in Base 7
(49, 63, 2402)-Net over F7 — Constructive and digital
Digital (49, 63, 2402)-net over F7, using
- t-expansion [i] based on digital (48, 63, 2402)-net over F7, using
- net defined by OOA [i] based on linear OOA(763, 2402, F7, 15, 15) (dual of [(2402, 15), 35967, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(763, 16815, F7, 15) (dual of [16815, 16752, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [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(751, 16807, F7, 12) (dual of [16807, 16756, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(72, 8, F7, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,7)), using
- extended Reed–Solomon code RSe(6,7) [i]
- Hamming code H(2,7) [i]
- algebraic-geometric code AG(F, Q+1P) with degQ = 3 and degPÂ =Â 2 [i] based on function field F/F7 with g(F) = 0 and N(F) ≥ 8, using the rational function field F7(x) [i]
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(763, 16815, F7, 15) (dual of [16815, 16752, 16]-code), using
- net defined by OOA [i] based on linear OOA(763, 2402, F7, 15, 15) (dual of [(2402, 15), 35967, 16]-NRT-code), using
(49, 63, 16820)-Net over F7 — Digital
Digital (49, 63, 16820)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(763, 16820, F7, 14) (dual of [16820, 16757, 15]-code), using
- construction XX applied to Ce(14) ⊂ Ce(11) ⊂ Ce(10) [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(751, 16807, F7, 12) (dual of [16807, 16756, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(71, 12, F7, 1) (dual of [12, 11, 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(11) ⊂ Ce(10) [i] based on
(49, 63, large)-Net in Base 7 — Upper bound on s
There is no (49, 63, large)-net in base 7, because
- 12 times m-reduction [i] would yield (49, 51, large)-net in base 7, but