Best Known (13, 13+4, s)-Nets in Base 9
(13, 13+4, 29530)-Net over F9 — Constructive and digital
Digital (13, 17, 29530)-net over F9, using
- net defined by OOA [i] based on linear OOA(917, 29530, F9, 4, 4) (dual of [(29530, 4), 118103, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(917, 29530, F9, 3, 4) (dual of [(29530, 3), 88573, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(917, 59060, F9, 4) (dual of [59060, 59043, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(1) [i] based on
- linear OA(916, 59049, F9, 4) (dual of [59049, 59033, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(96, 59049, F9, 2) (dual of [59049, 59043, 3]-code), using an extension Ce(1) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,1], and designed minimum distance d ≥ |I|+1 = 2 [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(3) ⊂ Ce(1) [i] based on
- OA 2-folding and stacking [i] based on linear OA(917, 59060, F9, 4) (dual of [59060, 59043, 5]-code), using
- appending kth column [i] based on linear OOA(917, 29530, F9, 3, 4) (dual of [(29530, 3), 88573, 5]-NRT-code), using
(13, 13+4, 59061)-Net over F9 — Digital
Digital (13, 17, 59061)-net over F9, using
- net defined by OOA [i] based on linear OOA(917, 59061, F9, 4, 4) (dual of [(59061, 4), 236227, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(917, 59061, F9, 3, 4) (dual of [(59061, 3), 177166, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(917, 59061, F9, 4) (dual of [59061, 59044, 5]-code), using
- construction X4 applied to Ce(3) ⊂ Ce(1) [i] based on
- linear OA(916, 59049, F9, 4) (dual of [59049, 59033, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(96, 59049, F9, 2) (dual of [59049, 59043, 3]-code), using an extension Ce(1) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,1], and designed minimum distance d ≥ |I|+1 = 2 [i]
- linear OA(911, 12, F9, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,9)), using
- dual of repetition code with length 12 [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
- construction X4 applied to Ce(3) ⊂ Ce(1) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(917, 59061, F9, 4) (dual of [59061, 59044, 5]-code), using
- appending kth column [i] based on linear OOA(917, 59061, F9, 3, 4) (dual of [(59061, 3), 177166, 5]-NRT-code), using
(13, 13+4, large)-Net in Base 9 — Upper bound on s
There is no (13, 17, large)-net in base 9, because
- 2 times m-reduction [i] would yield (13, 15, large)-net in base 9, but