Best Known (110−11, 110, s)-Nets in Base 4
(110−11, 110, 1677847)-Net over F4 — Constructive and digital
Digital (99, 110, 1677847)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (8, 13, 127)-net over F4, using
- net defined by OOA [i] based on linear OOA(413, 127, F4, 5, 5) (dual of [(127, 5), 622, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(413, 255, F4, 5) (dual of [255, 242, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- OOA 2-folding and stacking with additional row [i] based on linear OA(413, 255, F4, 5) (dual of [255, 242, 6]-code), using
- net defined by OOA [i] based on linear OOA(413, 127, F4, 5, 5) (dual of [(127, 5), 622, 6]-NRT-code), using
- digital (86, 97, 1677720)-net over F4, using
- net defined by OOA [i] based on linear OOA(497, 1677720, F4, 11, 11) (dual of [(1677720, 11), 18454823, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(497, 8388601, F4, 11) (dual of [8388601, 8388504, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(497, large, F4, 11) (dual of [large, large−97, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(497, large, F4, 11) (dual of [large, large−97, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(497, 8388601, F4, 11) (dual of [8388601, 8388504, 12]-code), using
- net defined by OOA [i] based on linear OOA(497, 1677720, F4, 11, 11) (dual of [(1677720, 11), 18454823, 12]-NRT-code), using
- digital (8, 13, 127)-net over F4, using
(110−11, 110, large)-Net over F4 — Digital
Digital (99, 110, large)-net over F4, using
- 47 times duplication [i] based on digital (92, 103, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4103, large, F4, 11) (dual of [large, large−103, 12]-code), using
- 6 times code embedding in larger space [i] based on linear OA(497, large, F4, 11) (dual of [large, large−97, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- 6 times code embedding in larger space [i] based on linear OA(497, large, F4, 11) (dual of [large, large−97, 12]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4103, large, F4, 11) (dual of [large, large−103, 12]-code), using
(110−11, 110, large)-Net in Base 4 — Upper bound on s
There is no (99, 110, large)-net in base 4, because
- 9 times m-reduction [i] would yield (99, 101, large)-net in base 4, but