Best Known (123, 123+11, s)-Nets in Base 2
(123, 123+11, 2097151)-Net over F2 — Constructive and digital
Digital (123, 134, 2097151)-net over F2, using
- net defined by OOA [i] based on linear OOA(2134, 2097151, F2, 15, 11) (dual of [(2097151, 15), 31457131, 12]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(2134, 4194303, F2, 3, 11) (dual of [(4194303, 3), 12582775, 12]-NRT-code), using
(123, 123+11, 2097407)-Net over F2 — Digital
Digital (123, 134, 2097407)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2134, 2097407, F2, 4, 11) (dual of [(2097407, 4), 8389494, 12]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(218, 257, F2, 4, 5) (dual of [(257, 4), 1010, 6]-NRT-code), using
- extracting embedded OOA [i] based on digital (13, 18, 257)-net over F2, using
- linear OOA(2116, 2097150, F2, 4, 11) (dual of [(2097150, 4), 8388484, 12]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2116, 8388600, F2, 11) (dual of [8388600, 8388484, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(2116, large, F2, 11) (dual of [large, large−116, 12]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- discarding factors / shortening the dual code based on linear OA(2116, large, F2, 11) (dual of [large, large−116, 12]-code), using
- OOA 4-folding [i] based on linear OA(2116, 8388600, F2, 11) (dual of [8388600, 8388484, 12]-code), using
- linear OOA(218, 257, F2, 4, 5) (dual of [(257, 4), 1010, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
(123, 123+11, large)-Net in Base 2 — Upper bound on s
There is no (123, 134, large)-net in base 2, because
- 9 times m-reduction [i] would yield (123, 125, large)-net in base 2, but