Best Known (78−18, 78, s)-Nets in Base 7
(78−18, 78, 1868)-Net over F7 — Constructive and digital
Digital (60, 78, 1868)-net over F7, using
- 72 times duplication [i] based on digital (58, 76, 1868)-net over F7, using
- net defined by OOA [i] based on linear OOA(776, 1868, F7, 18, 18) (dual of [(1868, 18), 33548, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(776, 16812, F7, 18) (dual of [16812, 16736, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(776, 16807, F7, 18) (dual of [16807, 16731, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(771, 16807, F7, 17) (dual of [16807, 16736, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(17) ⊂ Ce(16) [i] based on
- OA 9-folding and stacking [i] based on linear OA(776, 16812, F7, 18) (dual of [16812, 16736, 19]-code), using
- net defined by OOA [i] based on linear OOA(776, 1868, F7, 18, 18) (dual of [(1868, 18), 33548, 19]-NRT-code), using
(78−18, 78, 13216)-Net over F7 — Digital
Digital (60, 78, 13216)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(778, 13216, F7, 18) (dual of [13216, 13138, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(778, 16820, F7, 18) (dual of [16820, 16742, 19]-code), using
- construction XX applied to Ce(17) ⊂ Ce(15) ⊂ Ce(14) [i] based on
- linear OA(776, 16807, F7, 18) (dual of [16807, 16731, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(766, 16807, F7, 16) (dual of [16807, 16741, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- 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(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(17) ⊂ Ce(15) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(778, 16820, F7, 18) (dual of [16820, 16742, 19]-code), using
(78−18, 78, large)-Net in Base 7 — Upper bound on s
There is no (60, 78, large)-net in base 7, because
- 16 times m-reduction [i] would yield (60, 62, large)-net in base 7, but