Best Known (128−23, 128, s)-Nets in Base 4
(128−23, 128, 1492)-Net over F4 — Constructive and digital
Digital (105, 128, 1492)-net over F4, using
- 41 times duplication [i] based on digital (104, 127, 1492)-net over F4, using
- net defined by OOA [i] based on linear OOA(4127, 1492, F4, 23, 23) (dual of [(1492, 23), 34189, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4127, 16413, F4, 23) (dual of [16413, 16286, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4127, 16419, F4, 23) (dual of [16419, 16292, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- linear OA(4120, 16384, F4, 23) (dual of [16384, 16264, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(492, 16384, F4, 18) (dual of [16384, 16292, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(47, 35, F4, 4) (dual of [35, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4127, 16419, F4, 23) (dual of [16419, 16292, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4127, 16413, F4, 23) (dual of [16413, 16286, 24]-code), using
- net defined by OOA [i] based on linear OOA(4127, 1492, F4, 23, 23) (dual of [(1492, 23), 34189, 24]-NRT-code), using
(128−23, 128, 12643)-Net over F4 — Digital
Digital (105, 128, 12643)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4128, 12643, F4, 23) (dual of [12643, 12515, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4128, 16400, F4, 23) (dual of [16400, 16272, 24]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(4127, 16385, F4, 25) (dual of [16385, 16258, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(4113, 16385, F4, 21) (dual of [16385, 16272, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(41, 15, F4, 1) (dual of [15, 14, 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([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4128, 16400, F4, 23) (dual of [16400, 16272, 24]-code), using
(128−23, 128, large)-Net in Base 4 — Upper bound on s
There is no (105, 128, large)-net in base 4, because
- 21 times m-reduction [i] would yield (105, 107, large)-net in base 4, but