Best Known (82, 82+25, s)-Nets in Base 7
(82, 82+25, 1401)-Net over F7 — Constructive and digital
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
(82, 82+25, 12320)-Net over F7 — Digital
Digital (82, 107, 12320)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7107, 12320, F7, 25) (dual of [12320, 12213, 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
(82, 82+25, large)-Net in Base 7 — Upper bound on s
There is no (82, 107, large)-net in base 7, because
- 23 times m-reduction [i] would yield (82, 84, large)-net in base 7, but