Best Known (44−9, 44, s)-Nets in Base 2
(44−9, 44, 261)-Net over F2 — Constructive and digital
Digital (35, 44, 261)-net over F2, using
- 21 times duplication [i] based on digital (34, 43, 261)-net over F2, using
- net defined by OOA [i] based on linear OOA(243, 261, F2, 9, 9) (dual of [(261, 9), 2306, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(243, 261, F2, 8, 9) (dual of [(261, 8), 2045, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(243, 1045, F2, 9) (dual of [1045, 1002, 10]-code), using
- construction XX applied to C1 = C([1021,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([1021,6]) [i] based on
- linear OA(231, 1023, F2, 7) (dual of [1023, 992, 8]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,4}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(231, 1023, F2, 7) (dual of [1023, 992, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(241, 1023, F2, 9) (dual of [1023, 982, 10]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(221, 1023, F2, 5) (dual of [1023, 1002, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(21, 11, F2, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(21, 11, F2, 1) (dual of [11, 10, 2]-code) (see above)
- construction XX applied to C1 = C([1021,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([1021,6]) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(243, 1045, F2, 9) (dual of [1045, 1002, 10]-code), using
- appending kth column [i] based on linear OOA(243, 261, F2, 8, 9) (dual of [(261, 8), 2045, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(243, 261, F2, 9, 9) (dual of [(261, 9), 2306, 10]-NRT-code), using
(44−9, 44, 375)-Net over F2 — Digital
Digital (35, 44, 375)-net over F2, using
- net defined by OOA [i] based on linear OOA(244, 375, F2, 9, 9) (dual of [(375, 9), 3331, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(244, 375, F2, 8, 9) (dual of [(375, 8), 2956, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(244, 375, F2, 2, 9) (dual of [(375, 2), 706, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(244, 523, F2, 2, 9) (dual of [(523, 2), 1002, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(244, 1046, F2, 9) (dual of [1046, 1002, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(244, 1047, F2, 9) (dual of [1047, 1003, 10]-code), using
- adding a parity check bit [i] based on linear OA(243, 1046, F2, 8) (dual of [1046, 1003, 9]-code), using
- construction XX applied to C1 = C([1021,4]), C2 = C([1,6]), C3 = C1 + C2 = C([1,4]), and C∩ = C1 ∩ C2 = C([1021,6]) [i] based on
- linear OA(231, 1023, F2, 7) (dual of [1023, 992, 8]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,4}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(230, 1023, F2, 6) (dual of [1023, 993, 7]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(241, 1023, F2, 9) (dual of [1023, 982, 10]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(220, 1023, F2, 4) (dual of [1023, 1003, 5]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(21, 12, F2, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(21, 11, F2, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s (see above)
- construction XX applied to C1 = C([1021,4]), C2 = C([1,6]), C3 = C1 + C2 = C([1,4]), and C∩ = C1 ∩ C2 = C([1021,6]) [i] based on
- adding a parity check bit [i] based on linear OA(243, 1046, F2, 8) (dual of [1046, 1003, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(244, 1047, F2, 9) (dual of [1047, 1003, 10]-code), using
- OOA 2-folding [i] based on linear OA(244, 1046, F2, 9) (dual of [1046, 1002, 10]-code), using
- discarding factors / shortening the dual code based on linear OOA(244, 523, F2, 2, 9) (dual of [(523, 2), 1002, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(244, 375, F2, 2, 9) (dual of [(375, 2), 706, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(244, 375, F2, 8, 9) (dual of [(375, 8), 2956, 10]-NRT-code), using
(44−9, 44, 3806)-Net in Base 2 — Upper bound on s
There is no (35, 44, 3807)-net in base 2, because
- 1 times m-reduction [i] would yield (35, 43, 3807)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 8 802918 064693 > 243 [i]