Best Known (23−4, 23, s)-Nets in Base 9
(23−4, 23, 2391492)-Net over F9 — Constructive and digital
Digital (19, 23, 2391492)-net over F9, using
- net defined by OOA [i] based on linear OOA(923, 2391492, F9, 4, 4) (dual of [(2391492, 4), 9565945, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(923, 2391492, F9, 3, 4) (dual of [(2391492, 3), 7174453, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(923, 4782984, F9, 4) (dual of [4782984, 4782961, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(1) [i] based on
- linear OA(922, 4782969, F9, 4) (dual of [4782969, 4782947, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(98, 4782969, F9, 2) (dual of [4782969, 4782961, 3]-code), using an extension Ce(1) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,1], and designed minimum distance d ≥ |I|+1 = 2 [i]
- linear OA(91, 15, F9, 1) (dual of [15, 14, 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(923, 4782984, F9, 4) (dual of [4782984, 4782961, 5]-code), using
- appending kth column [i] based on linear OOA(923, 2391492, F9, 3, 4) (dual of [(2391492, 3), 7174453, 5]-NRT-code), using
(23−4, 23, 4782985)-Net over F9 — Digital
Digital (19, 23, 4782985)-net over F9, using
- net defined by OOA [i] based on linear OOA(923, 4782985, F9, 4, 4) (dual of [(4782985, 4), 19131917, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(923, 4782985, F9, 3, 4) (dual of [(4782985, 3), 14348932, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(923, 4782985, F9, 4) (dual of [4782985, 4782962, 5]-code), using
- construction X4 applied to Ce(3) ⊂ Ce(1) [i] based on
- linear OA(922, 4782969, F9, 4) (dual of [4782969, 4782947, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(98, 4782969, F9, 2) (dual of [4782969, 4782961, 3]-code), using an extension Ce(1) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,1], and designed minimum distance d ≥ |I|+1 = 2 [i]
- linear OA(915, 16, F9, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,9)), using
- dual of repetition code with length 16 [i]
- linear OA(91, 16, F9, 1) (dual of [16, 15, 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(923, 4782985, F9, 4) (dual of [4782985, 4782962, 5]-code), using
- appending kth column [i] based on linear OOA(923, 4782985, F9, 3, 4) (dual of [(4782985, 3), 14348932, 5]-NRT-code), using
(23−4, 23, large)-Net in Base 9 — Upper bound on s
There is no (19, 23, large)-net in base 9, because
- 2 times m-reduction [i] would yield (19, 21, large)-net in base 9, but