Best Known (207−32, 207, s)-Nets in Base 4
(207−32, 207, 4099)-Net over F4 — Constructive and digital
Digital (175, 207, 4099)-net over F4, using
- 41 times duplication [i] based on digital (174, 206, 4099)-net over F4, using
- t-expansion [i] based on digital (173, 206, 4099)-net over F4, using
- net defined by OOA [i] based on linear OOA(4206, 4099, F4, 33, 33) (dual of [(4099, 33), 135061, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4206, 65585, F4, 33) (dual of [65585, 65379, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4206, 65589, F4, 33) (dual of [65589, 65383, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(25) [i] based on
- linear OA(4193, 65536, F4, 33) (dual of [65536, 65343, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(4153, 65536, F4, 26) (dual of [65536, 65383, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(413, 53, F4, 6) (dual of [53, 40, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to Ce(32) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(4206, 65589, F4, 33) (dual of [65589, 65383, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4206, 65585, F4, 33) (dual of [65585, 65379, 34]-code), using
- net defined by OOA [i] based on linear OOA(4206, 4099, F4, 33, 33) (dual of [(4099, 33), 135061, 34]-NRT-code), using
- t-expansion [i] based on digital (173, 206, 4099)-net over F4, using
(207−32, 207, 54654)-Net over F4 — Digital
Digital (175, 207, 54654)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4207, 54654, F4, 32) (dual of [54654, 54447, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(4207, 65599, F4, 32) (dual of [65599, 65392, 33]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4206, 65598, F4, 32) (dual of [65598, 65392, 33]-code), using
- construction X applied to C([0,16]) ⊂ C([0,12]) [i] based on
- linear OA(4193, 65537, F4, 33) (dual of [65537, 65344, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(4145, 65537, F4, 25) (dual of [65537, 65392, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(413, 61, F4, 6) (dual of [61, 48, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to C([0,16]) ⊂ C([0,12]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4206, 65598, F4, 32) (dual of [65598, 65392, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(4207, 65599, F4, 32) (dual of [65599, 65392, 33]-code), using
(207−32, 207, large)-Net in Base 4 — Upper bound on s
There is no (175, 207, large)-net in base 4, because
- 30 times m-reduction [i] would yield (175, 177, large)-net in base 4, but