Best Known (80, 104, s)-Nets in Base 7
(80, 104, 1401)-Net over F7 — Constructive and digital
Digital (80, 104, 1401)-net over F7, using
- 73 times duplication [i] based on digital (77, 101, 1401)-net over F7, using
- net defined by OOA [i] based on linear OOA(7101, 1401, F7, 24, 24) (dual of [(1401, 24), 33523, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(7101, 16812, F7, 24) (dual of [16812, 16711, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(22) [i] based on
- linear OA(7101, 16807, F7, 24) (dual of [16807, 16706, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(796, 16807, F7, 23) (dual of [16807, 16711, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(23) ⊂ Ce(22) [i] based on
- OA 12-folding and stacking [i] based on linear OA(7101, 16812, F7, 24) (dual of [16812, 16711, 25]-code), using
- net defined by OOA [i] based on linear OOA(7101, 1401, F7, 24, 24) (dual of [(1401, 24), 33523, 25]-NRT-code), using
(80, 104, 13642)-Net over F7 — Digital
Digital (80, 104, 13642)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7104, 13642, F7, 24) (dual of [13642, 13538, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(7104, 16821, F7, 24) (dual of [16821, 16717, 25]-code), using
- construction XX applied to Ce(23) ⊂ Ce(21) ⊂ Ce(19) [i] based on
- linear OA(7101, 16807, F7, 24) (dual of [16807, 16706, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(791, 16807, F7, 22) (dual of [16807, 16716, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(786, 16807, F7, 20) (dual of [16807, 16721, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(71, 2, F7, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(23) ⊂ Ce(21) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(7104, 16821, F7, 24) (dual of [16821, 16717, 25]-code), using
(80, 104, large)-Net in Base 7 — Upper bound on s
There is no (80, 104, large)-net in base 7, because
- 22 times m-reduction [i] would yield (80, 82, large)-net in base 7, but