Best Known (81, 100, s)-Nets in Base 4
(81, 100, 1822)-Net over F4 — Constructive and digital
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
(81, 100, 8200)-Net over F4 — Digital
Digital (81, 100, 8200)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4100, 8200, F4, 2, 19) (dual of [(8200, 2), 16300, 20]-NRT-code), using
- OOA 2-folding [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
- OOA 2-folding [i] based on linear OA(4100, 16400, F4, 19) (dual of [16400, 16300, 20]-code), using
(81, 100, 5798151)-Net in Base 4 — Upper bound on s
There is no (81, 100, 5798152)-net in base 4, because
- 1 times m-reduction [i] would yield (81, 99, 5798152)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 401734 699514 949625 778643 313016 567241 077052 370048 829227 511776 > 499 [i]