MinT - Best Known (209, 242, s)-Nets in Base 2

Best Known (209, 242, s)-Nets in Base 2

(209, 242, 2048)-Net over F₂ — Constructive and digital

Digital (209, 242, 2048)-net over F₂, using

2¹ times duplication [i] based on digital (208, 241, 2048)-net over F₂, using
- net defined by OOA [i] based on linear OOA(2²⁴¹, 2048, F₂, 33, 33) (dual of [(2048, 33), 67343, 34]-NRT-code), using
  - OOA 16-folding and stacking with additional row [i] based on linear OA(2²⁴¹, 32769, F₂, 33) (dual of [32769, 32528, 34]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 32769Â | 2³⁰âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]

(209, 242, 5464)-Net over F₂ — Digital

Digital (209, 242, 5464)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁴², 5464, F₂, 6, 33) (dual of [(5464, 6), 32542, 34]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2²⁴², 32784, F₂, 33) (dual of [32784, 32542, 34]-code), using
  - constructionÂ X applied to C_e(32) âŠ‚ C_e(30) [i] based on
    1. linear OA(2²⁴¹, 32768, F₂, 33) (dual of [32768, 32527, 34]-code), using an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 2¹⁵âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]
    2. linear OA(2²²⁶, 32768, F₂, 31) (dual of [32768, 32542, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 2¹⁵âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
    3. linear OA(2¹, 16, F₂, 1) (dual of [16, 15, 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
        parity-check code [i]

(209, 242, 232680)-Net in Base 2 — Upper bound on s

There is no (209, 242, 232681)-net in base 2, because

1 times m-reduction [i] would yield (209, 241, 232681)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3 533921 359892 225637 661687 662153 695867 321235 036857 890821 727167 849933 419470 > 2²⁴¹ [i]