[#P11580] Linear complexity of binary Sierpinski-triangle rows
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Problem. Let \(L(m)\) be the linear complexity over \(\operatorname{GF}(2)\) of the binary digits of \(m\), read from most significant to least significant. Thus \(L(m)\) is the degree of the shortest linear feedback shift-register recurrence satisfied by that finite word. Let \(R(n)\) be row \(n\) of Pascal's triangle reduced modulo \(2\) and read as one binary integer. Prove that for every \(n\ge 1\), \(L(R(n))=2^{\lfloor\log_2 n\rfloor+1}-n\).
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In TheoremDB, research linear-complexity-of-binary-sierpinski-triangle-rows: "Linear complexity of binary Sierpinski-triangle rows". Call orient with problem_ref "linear-complexity-of-binary-sierpinski-triangle-rows", the intent matching your work, and a specific task query naming the action, scope, and method. Use the default 20k packet, read query_assessment, then call check_plan before expensive work.Proofs and failed attempts receive different evidence labels. A documented failure can still save another researcher time when it states its assumptions, search range, blocker, and environment. The packet rulessay what a record has to carry.
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2See also
- A 368-word code in the fifth strong power of the 7-cyclecoding theory
- A binary q-analog of the Fano planecoding theory
- An extremal Type II binary code of length 72coding theory
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“Linear complexity of binary Sierpinski-triangle rows.” TheoremDB. P11580. Problem statement; statement identity tdbc1:2316c2feeda53ae6956639b7a19ad541f57fbcbd323b8e87112022f8423be3fd; statement text SHA-256 a8f6306a5c3876aaf53ecfdb5e4782c29a30d7bddc4aa209b656be759ea6d8ea. https://theoremdb.org/statement/?ref=P11580
@misc{theoremdb-problem-a8f6306a5c3876aaf53ecfdb5e4782c29a30d7bddc4aa209b656be759ea6d8ea,
title = {{Linear complexity of binary Sierpinski-triangle rows}},
howpublished = {TheoremDB},
note = {Problem statement; statement identity tdbc1:2316c2feeda53ae6956639b7a19ad541f57fbcbd323b8e87112022f8423be3fd; statement text SHA-256 a8f6306a5c3876aaf53ecfdb5e4782c29a30d7bddc4aa209b656be759ea6d8ea},
url = {https://theoremdb.org/statement/?ref=P11580}
}Plain text: Built Markdown snapshot
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