MinT - Best Known (49−11, 49, s)-Nets in Base 16

Best Known (49−11, 49, s)-Nets in Base 16

(49−11, 49, 26335)-Net over F₁₆ — Constructive and digital

Digital (38, 49, 26335)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 7, 120)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁷, 120, F₁₆, 5, 5) (dual of [(120, 5), 593, 6]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OA(16⁷, 241, F₁₆, 5) (dual of [241, 234, 6]-code), using
      - a code by Danev and Olsson [i]
2. digital (31, 42, 26215)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁴², 26215, F₁₆, 11, 11) (dual of [(26215, 11), 288323, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(16⁴², 131076, F₁₆, 11) (dual of [131076, 131034, 12]-code), using
      - trace code [i] based on linear OA(256²¹, 65538, F₂₅₆, 11) (dual of [65538, 65517, 12]-code), using
        constructionÂ X applied to C_e(10) âŠ‚ C_e(9) [i] based on
        linear OA(256²¹, 65536, F₂₅₆, 11) (dual of [65536, 65515, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
        linear OA(256¹⁹, 65536, F₂₅₆, 10) (dual of [65536, 65517, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
        linear OA(256⁰, 2, F₂₅₆, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁰, s, F₂₅₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(49−11, 49, 52430)-Net in Base 16 — Constructive

(38, 49, 52430)-net in base 16, using

16¹ times duplication [i] based on (37, 48, 52430)-net in base 16, using
- base change [i] based on digital (21, 32, 52430)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64³², 52430, F₆₄, 11, 11) (dual of [(52430, 11), 576698, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(64³², 262151, F₆₄, 11) (dual of [262151, 262119, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(64³², 262152, F₆₄, 11) (dual of [262152, 262120, 12]-code), using
        constructionÂ X applied to C([0,5]) âŠ‚ C([0,4]) [i] based on
        linear OA(64³¹, 262145, F₆₄, 11) (dual of [262145, 262114, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 64⁶âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]
        linear OA(64²⁵, 262145, F₆₄, 9) (dual of [262145, 262120, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 64⁶âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]
        linear OA(64¹, 7, F₆₄, 1) (dual of [7, 6, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(64¹, s, F₆₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(49−11, 49, 239929)-Net over F₁₆ — Digital

Digital (38, 49, 239929)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(49−11, 49, large)-Net in Base 16 — Upper bound on s

There is no (38, 49, large)-net in base 16, because

9 times m-reduction [i] would yield (38, 40, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (49−11, 49, s)-Nets in Base 16

(49−11, 49, 26335)-Net over F16 — Constructive and digital

(49−11, 49, 52430)-Net in Base 16 — Constructive

(49−11, 49, 239929)-Net over F16 — Digital

(49−11, 49, large)-Net in Base 16 — Upper bound on s

(49−11, 49, 26335)-Net over F₁₆ — Constructive and digital

(49−11, 49, 239929)-Net over F₁₆ — Digital