Best Known (244−28, 244, s)-Nets in Base 4
(244−28, 244, 299597)-Net over F4 — Constructive and digital
Digital (216, 244, 299597)-net over F4, using
- 41 times duplication [i] based on digital (215, 243, 299597)-net over F4, using
- t-expansion [i] based on digital (214, 243, 299597)-net over F4, using
- net defined by OOA [i] based on linear OOA(4243, 299597, F4, 29, 29) (dual of [(299597, 29), 8688070, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4243, 4194359, F4, 29) (dual of [4194359, 4194116, 30]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4242, 4194358, F4, 29) (dual of [4194358, 4194116, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- linear OA(4232, 4194304, F4, 29) (dual of [4194304, 4194072, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4188, 4194304, F4, 23) (dual of [4194304, 4194116, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(410, 54, F4, 5) (dual of [54, 44, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4242, 4194358, F4, 29) (dual of [4194358, 4194116, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4243, 4194359, F4, 29) (dual of [4194359, 4194116, 30]-code), using
- net defined by OOA [i] based on linear OOA(4243, 299597, F4, 29, 29) (dual of [(299597, 29), 8688070, 30]-NRT-code), using
- t-expansion [i] based on digital (214, 243, 299597)-net over F4, using
(244−28, 244, 2097185)-Net over F4 — Digital
Digital (216, 244, 2097185)-net over F4, using
- 41 times duplication [i] based on digital (215, 243, 2097185)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4243, 2097185, F4, 2, 28) (dual of [(2097185, 2), 4194127, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4243, 4194370, F4, 28) (dual of [4194370, 4194127, 29]-code), using
- construction X applied to Ce(28) ⊂ Ce(21) [i] based on
- linear OA(4232, 4194304, F4, 29) (dual of [4194304, 4194072, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4177, 4194304, F4, 22) (dual of [4194304, 4194127, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(411, 66, F4, 5) (dual of [66, 55, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(411, 68, F4, 5) (dual of [68, 57, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(410, 64, F4, 5) (dual of [64, 54, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(47, 64, F4, 3) (dual of [64, 57, 4]-code or 64-cap in PG(6,4)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(41, 4, F4, 1) (dual of [4, 3, 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 Ce(4) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(411, 68, F4, 5) (dual of [68, 57, 6]-code), using
- construction X applied to Ce(28) ⊂ Ce(21) [i] based on
- OOA 2-folding [i] based on linear OA(4243, 4194370, F4, 28) (dual of [4194370, 4194127, 29]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4243, 2097185, F4, 2, 28) (dual of [(2097185, 2), 4194127, 29]-NRT-code), using
(244−28, 244, large)-Net in Base 4 — Upper bound on s
There is no (216, 244, large)-net in base 4, because
- 26 times m-reduction [i] would yield (216, 218, large)-net in base 4, but