Best Known (129−28, 129, s)-Nets in Base 9
(129−28, 129, 4220)-Net over F9 — Constructive and digital
Digital (101, 129, 4220)-net over F9, using
- 91 times duplication [i] based on digital (100, 128, 4220)-net over F9, using
- net defined by OOA [i] based on linear OOA(9128, 4220, F9, 28, 28) (dual of [(4220, 28), 118032, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(9128, 59080, F9, 28) (dual of [59080, 58952, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(9128, 59081, F9, 28) (dual of [59081, 58953, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- linear OA(9121, 59049, F9, 28) (dual of [59049, 58928, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(996, 59049, F9, 22) (dual of [59049, 58953, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(97, 32, F9, 5) (dual of [32, 25, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(9128, 59081, F9, 28) (dual of [59081, 58953, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(9128, 59080, F9, 28) (dual of [59080, 58952, 29]-code), using
- net defined by OOA [i] based on linear OOA(9128, 4220, F9, 28, 28) (dual of [(4220, 28), 118032, 29]-NRT-code), using
(129−28, 129, 59083)-Net over F9 — Digital
Digital (101, 129, 59083)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9129, 59083, F9, 28) (dual of [59083, 58954, 29]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(9128, 59081, F9, 28) (dual of [59081, 58953, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- linear OA(9121, 59049, F9, 28) (dual of [59049, 58928, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(996, 59049, F9, 22) (dual of [59049, 58953, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(97, 32, F9, 5) (dual of [32, 25, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- linear OA(9128, 59082, F9, 27) (dual of [59082, 58954, 28]-code), using Gilbert–Varšamov bound and bm = 9128 > Vbs−1(k−1) = 8 511359 759487 072952 950387 528848 414544 205046 889042 469099 155568 727971 778807 390178 300592 152375 725337 432808 771705 837269 933897 [i]
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(9128, 59081, F9, 28) (dual of [59081, 58953, 29]-code), using
- construction X with Varšamov bound [i] based on
(129−28, 129, large)-Net in Base 9 — Upper bound on s
There is no (101, 129, large)-net in base 9, because
- 26 times m-reduction [i] would yield (101, 103, large)-net in base 9, but