Best Known (49, 49+9, s)-Nets in Base 9
(49, 49+9, 1195746)-Net over F9 — Constructive and digital
Digital (49, 58, 1195746)-net over F9, using
- net defined by OOA [i] based on linear OOA(958, 1195746, F9, 9, 9) (dual of [(1195746, 9), 10761656, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(958, 4782985, F9, 9) (dual of [4782985, 4782927, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(957, 4782970, F9, 9) (dual of [4782970, 4782913, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 914−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(943, 4782970, F9, 7) (dual of [4782970, 4782927, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 914−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [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 C([0,4]) ⊂ C([0,3]) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(958, 4782985, F9, 9) (dual of [4782985, 4782927, 10]-code), using
(49, 49+9, 4782986)-Net over F9 — Digital
Digital (49, 58, 4782986)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(958, 4782986, F9, 9) (dual of [4782986, 4782928, 10]-code), using
- construction X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(957, 4782970, F9, 9) (dual of [4782970, 4782913, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 914−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(943, 4782970, F9, 7) (dual of [4782970, 4782927, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 914−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [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 C([0,4]) ⊂ C([0,3]) [i] based on
(49, 49+9, large)-Net in Base 9 — Upper bound on s
There is no (49, 58, large)-net in base 9, because
- 7 times m-reduction [i] would yield (49, 51, large)-net in base 9, but