Best Known (122, 122+28, s)-Nets in Base 9
(122, 122+28, 37961)-Net over F9 — Constructive and digital
Digital (122, 150, 37961)-net over F9, using
- 92 times duplication [i] based on digital (120, 148, 37961)-net over F9, using
- net defined by OOA [i] based on linear OOA(9148, 37961, F9, 28, 28) (dual of [(37961, 28), 1062760, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(9148, 531454, F9, 28) (dual of [531454, 531306, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(9148, 531456, F9, 28) (dual of [531456, 531308, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(24) [i] based on
- linear OA(9145, 531441, F9, 28) (dual of [531441, 531296, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(9133, 531441, F9, 25) (dual of [531441, 531308, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(93, 15, F9, 2) (dual of [15, 12, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 92−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- construction X applied to Ce(27) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(9148, 531456, F9, 28) (dual of [531456, 531308, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(9148, 531454, F9, 28) (dual of [531454, 531306, 29]-code), using
- net defined by OOA [i] based on linear OOA(9148, 37961, F9, 28, 28) (dual of [(37961, 28), 1062760, 29]-NRT-code), using
(122, 122+28, 387909)-Net over F9 — Digital
Digital (122, 150, 387909)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9150, 387909, F9, 28) (dual of [387909, 387759, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(9150, 531465, F9, 28) (dual of [531465, 531315, 29]-code), using
- construction XX applied to Ce(27) ⊂ Ce(23) ⊂ Ce(22) [i] based on
- linear OA(9145, 531441, F9, 28) (dual of [531441, 531296, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(9127, 531441, F9, 24) (dual of [531441, 531314, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(9121, 531441, F9, 23) (dual of [531441, 531320, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(94, 23, F9, 3) (dual of [23, 19, 4]-code or 23-cap in PG(3,9)), 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(27) ⊂ Ce(23) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(9150, 531465, F9, 28) (dual of [531465, 531315, 29]-code), using
(122, 122+28, large)-Net in Base 9 — Upper bound on s
There is no (122, 150, large)-net in base 9, because
- 26 times m-reduction [i] would yield (122, 124, large)-net in base 9, but