Best Known (117−22, 117, s)-Nets in Base 9
(117−22, 117, 48314)-Net over F9 — Constructive and digital
Digital (95, 117, 48314)-net over F9, using
- 91 times duplication [i] based on digital (94, 116, 48314)-net over F9, using
- net defined by OOA [i] based on linear OOA(9116, 48314, F9, 22, 22) (dual of [(48314, 22), 1062792, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(9116, 531454, F9, 22) (dual of [531454, 531338, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- linear OA(9115, 531441, F9, 22) (dual of [531441, 531326, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(9103, 531441, F9, 20) (dual of [531441, 531338, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(91, 13, F9, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- OA 11-folding and stacking [i] based on linear OA(9116, 531454, F9, 22) (dual of [531454, 531338, 23]-code), using
- net defined by OOA [i] based on linear OOA(9116, 48314, F9, 22, 22) (dual of [(48314, 22), 1062792, 23]-NRT-code), using
(117−22, 117, 355475)-Net over F9 — Digital
Digital (95, 117, 355475)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9117, 355475, F9, 22) (dual of [355475, 355358, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(9117, 531456, F9, 22) (dual of [531456, 531339, 23]-code), using
- construction XX applied to Ce(21) ⊂ Ce(19) ⊂ Ce(18) [i] based on
- linear OA(9115, 531441, F9, 22) (dual of [531441, 531326, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(9103, 531441, F9, 20) (dual of [531441, 531338, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(997, 531441, F9, 19) (dual of [531441, 531344, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(91, 14, F9, 1) (dual of [14, 13, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(21) ⊂ Ce(19) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(9117, 531456, F9, 22) (dual of [531456, 531339, 23]-code), using
(117−22, 117, large)-Net in Base 9 — Upper bound on s
There is no (95, 117, large)-net in base 9, because
- 20 times m-reduction [i] would yield (95, 97, large)-net in base 9, but