Best Known (48, 59, s)-Nets in Base 4
(48, 59, 3280)-Net over F4 — Constructive and digital
Digital (48, 59, 3280)-net over F4, using
- 41 times duplication [i] based on digital (47, 58, 3280)-net over F4, using
- net defined by OOA [i] based on linear OOA(458, 3280, F4, 11, 11) (dual of [(3280, 11), 36022, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(458, 16401, F4, 11) (dual of [16401, 16343, 12]-code), using
- construction X4 applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(457, 16385, F4, 11) (dual of [16385, 16328, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(443, 16385, F4, 9) (dual of [16385, 16342, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(415, 16, F4, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,4)), using
- dual of repetition code with length 16 [i]
- linear OA(41, 16, F4, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to C([0,5]) ⊂ C([0,4]) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(458, 16401, F4, 11) (dual of [16401, 16343, 12]-code), using
- net defined by OOA [i] based on linear OOA(458, 3280, F4, 11, 11) (dual of [(3280, 11), 36022, 12]-NRT-code), using
(48, 59, 10479)-Net over F4 — Digital
Digital (48, 59, 10479)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(459, 10479, F4, 11) (dual of [10479, 10420, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(459, 16402, F4, 11) (dual of [16402, 16343, 12]-code), using
- 1 times code embedding in larger space [i] based on linear OA(458, 16401, F4, 11) (dual of [16401, 16343, 12]-code), using
- construction X4 applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(457, 16385, F4, 11) (dual of [16385, 16328, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(443, 16385, F4, 9) (dual of [16385, 16342, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(415, 16, F4, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,4)), using
- dual of repetition code with length 16 [i]
- linear OA(41, 16, F4, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to C([0,5]) ⊂ C([0,4]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(458, 16401, F4, 11) (dual of [16401, 16343, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(459, 16402, F4, 11) (dual of [16402, 16343, 12]-code), using
(48, 59, 8367788)-Net in Base 4 — Upper bound on s
There is no (48, 59, 8367789)-net in base 4, because
- 1 times m-reduction [i] would yield (48, 58, 8367789)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 83076 766008 303343 429077 762955 194100 > 458 [i]