Best Known (83, 108, s)-Nets in Base 7
(83, 108, 1401)-Net over F7 — Constructive and digital
Digital (83, 108, 1401)-net over F7, using
- 71 times duplication [i] based on digital (82, 107, 1401)-net over F7, using
- net defined by OOA [i] based on linear OOA(7107, 1401, F7, 25, 25) (dual of [(1401, 25), 34918, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(7107, 16813, F7, 25) (dual of [16813, 16706, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(7107, 16818, F7, 25) (dual of [16818, 16711, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(7106, 16807, F7, 25) (dual of [16807, 16701, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(71, 11, F7, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(7107, 16818, F7, 25) (dual of [16818, 16711, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(7107, 16813, F7, 25) (dual of [16813, 16706, 26]-code), using
- net defined by OOA [i] based on linear OOA(7107, 1401, F7, 25, 25) (dual of [(1401, 25), 34918, 26]-NRT-code), using
(83, 108, 13409)-Net over F7 — Digital
Digital (83, 108, 13409)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7108, 13409, F7, 25) (dual of [13409, 13301, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(7108, 16820, F7, 25) (dual of [16820, 16712, 26]-code), using
- construction XX applied to Ce(24) ⊂ Ce(22) ⊂ Ce(21) [i] based on
- linear OA(7106, 16807, F7, 25) (dual of [16807, 16701, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(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(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(24) ⊂ Ce(22) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(7108, 16820, F7, 25) (dual of [16820, 16712, 26]-code), using
(83, 108, large)-Net in Base 7 — Upper bound on s
There is no (83, 108, large)-net in base 7, because
- 23 times m-reduction [i] would yield (83, 85, large)-net in base 7, but