Best Known (44−7, 44, s)-Nets in Base 9
(44−7, 44, 1594328)-Net over F9 — Constructive and digital
Digital (37, 44, 1594328)-net over F9, using
- net defined by OOA [i] based on linear OOA(944, 1594328, F9, 7, 7) (dual of [(1594328, 7), 11160252, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(944, 4782985, F9, 7) (dual of [4782985, 4782941, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- 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(929, 4782970, F9, 5) (dual of [4782970, 4782941, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 914−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (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([0,3]) ⊂ C([0,2]) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(944, 4782985, F9, 7) (dual of [4782985, 4782941, 8]-code), using
(44−7, 44, 4782986)-Net over F9 — Digital
Digital (37, 44, 4782986)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(944, 4782986, F9, 7) (dual of [4782986, 4782942, 8]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- 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(929, 4782970, F9, 5) (dual of [4782970, 4782941, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 914−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (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([0,3]) ⊂ C([0,2]) [i] based on
(44−7, 44, large)-Net in Base 9 — Upper bound on s
There is no (37, 44, large)-net in base 9, because
- 5 times m-reduction [i] would yield (37, 39, large)-net in base 9, but