Best Known (68−11, 68, s)-Nets in Base 4
(68−11, 68, 13111)-Net over F4 — Constructive and digital
Digital (57, 68, 13111)-net over F4, using
- 41 times duplication [i] based on digital (56, 67, 13111)-net over F4, using
- net defined by OOA [i] based on linear OOA(467, 13111, F4, 11, 11) (dual of [(13111, 11), 144154, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(467, 65556, F4, 11) (dual of [65556, 65489, 12]-code), using
- 1 times code embedding in larger space [i] based on linear OA(466, 65555, F4, 11) (dual of [65555, 65489, 12]-code), using
- construction X4 applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(465, 65537, F4, 11) (dual of [65537, 65472, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(449, 65537, F4, 9) (dual of [65537, 65488, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(417, 18, F4, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,4)), using
- dual of repetition code with length 18 [i]
- linear OA(41, 18, F4, 1) (dual of [18, 17, 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(466, 65555, F4, 11) (dual of [65555, 65489, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(467, 65556, F4, 11) (dual of [65556, 65489, 12]-code), using
- net defined by OOA [i] based on linear OOA(467, 13111, F4, 11, 11) (dual of [(13111, 11), 144154, 12]-NRT-code), using
(68−11, 68, 41934)-Net over F4 — Digital
Digital (57, 68, 41934)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(468, 41934, F4, 11) (dual of [41934, 41866, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(468, 65557, F4, 11) (dual of [65557, 65489, 12]-code), using
- 2 times code embedding in larger space [i] based on linear OA(466, 65555, F4, 11) (dual of [65555, 65489, 12]-code), using
- construction X4 applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(465, 65537, F4, 11) (dual of [65537, 65472, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(449, 65537, F4, 9) (dual of [65537, 65488, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(417, 18, F4, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,4)), using
- dual of repetition code with length 18 [i]
- linear OA(41, 18, F4, 1) (dual of [18, 17, 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
- 2 times code embedding in larger space [i] based on linear OA(466, 65555, F4, 11) (dual of [65555, 65489, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(468, 65557, F4, 11) (dual of [65557, 65489, 12]-code), using
(68−11, 68, large)-Net in Base 4 — Upper bound on s
There is no (57, 68, large)-net in base 4, because
- 9 times m-reduction [i] would yield (57, 59, large)-net in base 4, but