Best Known (114−11, 114, s)-Nets in Base 4
(114−11, 114, 1678235)-Net over F4 — Constructive and digital
Digital (103, 114, 1678235)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (12, 17, 515)-net over F4, using
- net defined by OOA [i] based on linear OOA(417, 515, F4, 5, 5) (dual of [(515, 5), 2558, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(417, 1031, F4, 5) (dual of [1031, 1014, 6]-code), using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(416, 1024, F4, 5) (dual of [1024, 1008, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(411, 1024, F4, 3) (dual of [1024, 1013, 4]-code or 1024-cap in PG(10,4)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(46, 7, F4, 6) (dual of [7, 1, 7]-code or 7-arc in PG(5,4)), using
- dual of repetition code with length 7 [i]
- linear OA(41, 7, F4, 1) (dual of [7, 6, 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 Ce(4) ⊂ Ce(2) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(417, 1031, F4, 5) (dual of [1031, 1014, 6]-code), using
- net defined by OOA [i] based on linear OOA(417, 515, F4, 5, 5) (dual of [(515, 5), 2558, 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 (12, 17, 515)-net over F4, using
(114−11, 114, large)-Net over F4 — Digital
Digital (103, 114, large)-net over F4, using
- t-expansion [i] based on digital (102, 114, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4114, large, F4, 12) (dual of [large, large−114, 13]-code), using
- 6 times code embedding in larger space [i] based on linear OA(4108, large, F4, 12) (dual of [large, large−108, 13]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 6 times code embedding in larger space [i] based on linear OA(4108, large, F4, 12) (dual of [large, large−108, 13]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4114, large, F4, 12) (dual of [large, large−114, 13]-code), using
(114−11, 114, large)-Net in Base 4 — Upper bound on s
There is no (103, 114, large)-net in base 4, because
- 9 times m-reduction [i] would yield (103, 105, large)-net in base 4, but