Best Known (109, 120, s)-Nets in Base 4
(109, 120, 1685916)-Net over F4 — Constructive and digital
Digital (109, 120, 1685916)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (18, 23, 8196)-net over F4, using
- net defined by OOA [i] based on linear OOA(423, 8196, F4, 5, 5) (dual of [(8196, 5), 40957, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(423, 16393, F4, 5) (dual of [16393, 16370, 6]-code), using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(422, 16384, F4, 5) (dual of [16384, 16362, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(415, 16384, F4, 3) (dual of [16384, 16369, 4]-code or 16384-cap in PG(14,4)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(48, 9, F4, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,4)), using
- dual of repetition code with length 9 [i]
- linear OA(41, 9, F4, 1) (dual of [9, 8, 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(423, 16393, F4, 5) (dual of [16393, 16370, 6]-code), using
- net defined by OOA [i] based on linear OOA(423, 8196, F4, 5, 5) (dual of [(8196, 5), 40957, 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 (18, 23, 8196)-net over F4, using
(109, 120, large)-Net over F4 — Digital
Digital (109, 120, large)-net over F4, using
- 46 times duplication [i] based on 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
- t-expansion [i] based on digital (102, 114, large)-net over F4, using
(109, 120, large)-Net in Base 4 — Upper bound on s
There is no (109, 120, large)-net in base 4, because
- 9 times m-reduction [i] would yield (109, 111, large)-net in base 4, but