Best Known (89, 89+15, s)-Nets in Base 3
(89, 89+15, 8439)-Net over F3 — Constructive and digital
Digital (89, 104, 8439)-net over F3, using
- net defined by OOA [i] based on linear OOA(3104, 8439, F3, 15, 15) (dual of [(8439, 15), 126481, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3104, 59074, F3, 15) (dual of [59074, 58970, 16]-code), using
- 2 times code embedding in larger space [i] based on linear OA(3102, 59072, F3, 15) (dual of [59072, 58970, 16]-code), using
- construction X4 applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(3101, 59050, F3, 15) (dual of [59050, 58949, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(381, 59050, F3, 13) (dual of [59050, 58969, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(321, 22, F3, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,3)), using
- dual of repetition code with length 22 [i]
- linear OA(31, 22, F3, 1) (dual of [22, 21, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to C([0,7]) ⊂ C([0,6]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(3102, 59072, F3, 15) (dual of [59072, 58970, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3104, 59074, F3, 15) (dual of [59074, 58970, 16]-code), using
(89, 89+15, 29537)-Net over F3 — Digital
Digital (89, 104, 29537)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3104, 29537, F3, 2, 15) (dual of [(29537, 2), 58970, 16]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3102, 29536, F3, 2, 15) (dual of [(29536, 2), 58970, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3102, 59072, F3, 15) (dual of [59072, 58970, 16]-code), using
- construction X4 applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(3101, 59050, F3, 15) (dual of [59050, 58949, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(381, 59050, F3, 13) (dual of [59050, 58969, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(321, 22, F3, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,3)), using
- dual of repetition code with length 22 [i]
- linear OA(31, 22, F3, 1) (dual of [22, 21, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to C([0,7]) ⊂ C([0,6]) [i] based on
- OOA 2-folding [i] based on linear OA(3102, 59072, F3, 15) (dual of [59072, 58970, 16]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3102, 29536, F3, 2, 15) (dual of [(29536, 2), 58970, 16]-NRT-code), using
(89, 89+15, large)-Net in Base 3 — Upper bound on s
There is no (89, 104, large)-net in base 3, because
- 13 times m-reduction [i] would yield (89, 91, large)-net in base 3, but