Best Known (104−23, 104, s)-Nets in Base 7
(104−23, 104, 1530)-Net over F7 — Constructive and digital
Digital (81, 104, 1530)-net over F7, using
- 72 times duplication [i] based on digital (79, 102, 1530)-net over F7, using
- net defined by OOA [i] based on linear OOA(7102, 1530, F7, 23, 23) (dual of [(1530, 23), 35088, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(7102, 16831, F7, 23) (dual of [16831, 16729, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(7102, 16833, F7, 23) (dual of [16833, 16731, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- 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(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(76, 26, F7, 4) (dual of [26, 20, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(76, 42, F7, 4) (dual of [42, 36, 5]-code), using
- 1 times truncation [i] based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(76, 42, F7, 4) (dual of [42, 36, 5]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(7102, 16833, F7, 23) (dual of [16833, 16731, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(7102, 16831, F7, 23) (dual of [16831, 16729, 24]-code), using
- net defined by OOA [i] based on linear OOA(7102, 1530, F7, 23, 23) (dual of [(1530, 23), 35088, 24]-NRT-code), using
(104−23, 104, 16841)-Net over F7 — Digital
Digital (81, 104, 16841)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7104, 16841, F7, 23) (dual of [16841, 16737, 24]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(7103, 16839, F7, 23) (dual of [16839, 16736, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- 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(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(77, 32, F7, 5) (dual of [32, 25, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- linear OA(7103, 16840, F7, 22) (dual of [16840, 16737, 23]-code), using Gilbert–Varšamov bound and bm = 7103 > Vbs−1(k−1) = 23 996431 312305 172653 319647 709054 911929 803346 902005 309521 315644 523170 661060 501274 527479 [i]
- linear OA(70, 1, F7, 0) (dual of [1, 1, 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
- linear OA(7103, 16839, F7, 23) (dual of [16839, 16736, 24]-code), using
- construction X with Varšamov bound [i] based on
(104−23, 104, large)-Net in Base 7 — Upper bound on s
There is no (81, 104, large)-net in base 7, because
- 21 times m-reduction [i] would yield (81, 83, large)-net in base 7, but