Best Known (30−5, 30, s)-Nets in Base 9
(30−5, 30, 2391492)-Net over F9 — Constructive and digital
Digital (25, 30, 2391492)-net over F9, using
- net defined by OOA [i] based on linear OOA(930, 2391492, F9, 5, 5) (dual of [(2391492, 5), 11957430, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(930, 4782985, F9, 5) (dual of [4782985, 4782955, 6]-code), using
- construction X applied to C([0,2]) ⊂ C([0,1]) [i] based on
- 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, 4782970, F9, 3) (dual of [4782970, 4782955, 4]-code or 4782970-cap in PG(14,9)), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 914−1, defining interval I = [0,1], and minimum distance d ≥ |{−1,0,1}|+1 = 4 (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([0,2]) ⊂ C([0,1]) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(930, 4782985, F9, 5) (dual of [4782985, 4782955, 6]-code), using
(30−5, 30, 4782986)-Net over F9 — Digital
Digital (25, 30, 4782986)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(930, 4782986, F9, 5) (dual of [4782986, 4782956, 6]-code), using
- construction X4 applied to C([0,2]) ⊂ C([0,1]) [i] based on
- 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, 4782970, F9, 3) (dual of [4782970, 4782955, 4]-code or 4782970-cap in PG(14,9)), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 914−1, defining interval I = [0,1], and minimum distance d ≥ |{−1,0,1}|+1 = 4 (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([0,2]) ⊂ C([0,1]) [i] based on
(30−5, 30, large)-Net in Base 9 — Upper bound on s
There is no (25, 30, large)-net in base 9, because
- 3 times m-reduction [i] would yield (25, 27, large)-net in base 9, but