Best Known (138−31, 138, s)-Nets in Base 9
(138−31, 138, 3937)-Net over F9 — Constructive and digital
Digital (107, 138, 3937)-net over F9, using
- 91 times duplication [i] based on digital (106, 137, 3937)-net over F9, using
- net defined by OOA [i] based on linear OOA(9137, 3937, F9, 31, 31) (dual of [(3937, 31), 121910, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(9137, 59056, F9, 31) (dual of [59056, 58919, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(9137, 59060, F9, 31) (dual of [59060, 58923, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(9136, 59049, F9, 31) (dual of [59049, 58913, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(9126, 59049, F9, 29) (dual of [59049, 58923, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(91, 11, F9, 1) (dual of [11, 10, 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(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(9137, 59060, F9, 31) (dual of [59060, 58923, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(9137, 59056, F9, 31) (dual of [59056, 58919, 32]-code), using
- net defined by OOA [i] based on linear OOA(9137, 3937, F9, 31, 31) (dual of [(3937, 31), 121910, 32]-NRT-code), using
(138−31, 138, 46973)-Net over F9 — Digital
Digital (107, 138, 46973)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9138, 46973, F9, 31) (dual of [46973, 46835, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(9138, 59062, F9, 31) (dual of [59062, 58924, 32]-code), using
- construction XX applied to Ce(30) ⊂ Ce(28) ⊂ Ce(27) [i] based on
- linear OA(9136, 59049, F9, 31) (dual of [59049, 58913, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(9126, 59049, F9, 29) (dual of [59049, 58923, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 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(91, 12, F9, 1) (dual of [12, 11, 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(30) ⊂ Ce(28) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(9138, 59062, F9, 31) (dual of [59062, 58924, 32]-code), using
(138−31, 138, large)-Net in Base 9 — Upper bound on s
There is no (107, 138, large)-net in base 9, because
- 29 times m-reduction [i] would yield (107, 109, large)-net in base 9, but