Best Known (82, 101, s)-Nets in Base 4
(82, 101, 1822)-Net over F4 — Constructive and digital
Digital (82, 101, 1822)-net over F4, using
- 41 times duplication [i] based on digital (81, 100, 1822)-net over F4, using
- net defined by OOA [i] based on linear OOA(4100, 1822, F4, 19, 19) (dual of [(1822, 19), 34518, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4100, 16399, F4, 19) (dual of [16399, 16299, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4100, 16400, F4, 19) (dual of [16400, 16300, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(499, 16385, F4, 19) (dual of [16385, 16286, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(485, 16385, F4, 17) (dual of [16385, 16300, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(41, 15, F4, 1) (dual of [15, 14, 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 X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4100, 16400, F4, 19) (dual of [16400, 16300, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4100, 16399, F4, 19) (dual of [16399, 16299, 20]-code), using
- net defined by OOA [i] based on linear OOA(4100, 1822, F4, 19, 19) (dual of [(1822, 19), 34518, 20]-NRT-code), using
(82, 101, 8311)-Net over F4 — Digital
Digital (82, 101, 8311)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4101, 8311, F4, 19) (dual of [8311, 8210, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4101, 16401, F4, 19) (dual of [16401, 16300, 20]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4100, 16400, F4, 19) (dual of [16400, 16300, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(499, 16385, F4, 19) (dual of [16385, 16286, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(485, 16385, F4, 17) (dual of [16385, 16300, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(41, 15, F4, 1) (dual of [15, 14, 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 X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4100, 16400, F4, 19) (dual of [16400, 16300, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4101, 16401, F4, 19) (dual of [16401, 16300, 20]-code), using
(82, 101, 6763713)-Net in Base 4 — Upper bound on s
There is no (82, 101, 6763714)-net in base 4, because
- 1 times m-reduction [i] would yield (82, 100, 6763714)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1 606939 119696 706430 506219 991064 105014 161664 514342 947228 162200 > 4100 [i]