Best Known (138, 138+19, s)-Nets in Base 4
(138, 138+19, 466036)-Net over F4 — Constructive and digital
Digital (138, 157, 466036)-net over F4, using
- 41 times duplication [i] based on digital (137, 156, 466036)-net over F4, using
- net defined by OOA [i] based on linear OOA(4156, 466036, F4, 19, 19) (dual of [(466036, 19), 8854528, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4156, 4194325, F4, 19) (dual of [4194325, 4194169, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4156, 4194328, F4, 19) (dual of [4194328, 4194172, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(4155, 4194305, F4, 19) (dual of [4194305, 4194150, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(4133, 4194305, F4, 17) (dual of [4194305, 4194172, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(41, 23, F4, 1) (dual of [23, 22, 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 C([0,9]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4156, 4194328, F4, 19) (dual of [4194328, 4194172, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4156, 4194325, F4, 19) (dual of [4194325, 4194169, 20]-code), using
- net defined by OOA [i] based on linear OOA(4156, 466036, F4, 19, 19) (dual of [(466036, 19), 8854528, 20]-NRT-code), using
(138, 138+19, 1541240)-Net over F4 — Digital
Digital (138, 157, 1541240)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4157, 1541240, F4, 2, 19) (dual of [(1541240, 2), 3082323, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4157, 2097165, F4, 2, 19) (dual of [(2097165, 2), 4194173, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4157, 4194330, F4, 19) (dual of [4194330, 4194173, 20]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4156, 4194329, F4, 19) (dual of [4194329, 4194173, 20]-code), using
- construction X4 applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(4155, 4194305, F4, 19) (dual of [4194305, 4194150, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(4133, 4194305, F4, 17) (dual of [4194305, 4194172, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(423, 24, F4, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,4)), using
- dual of repetition code with length 24 [i]
- linear OA(41, 24, F4, 1) (dual of [24, 23, 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 C([0,9]) ⊂ C([0,8]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4156, 4194329, F4, 19) (dual of [4194329, 4194173, 20]-code), using
- OOA 2-folding [i] based on linear OA(4157, 4194330, F4, 19) (dual of [4194330, 4194173, 20]-code), using
- discarding factors / shortening the dual code based on linear OOA(4157, 2097165, F4, 2, 19) (dual of [(2097165, 2), 4194173, 20]-NRT-code), using
(138, 138+19, large)-Net in Base 4 — Upper bound on s
There is no (138, 157, large)-net in base 4, because
- 17 times m-reduction [i] would yield (138, 140, large)-net in base 4, but