Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The trapping redundancy of a linear code is the number of rows of a smallest parity-check matrix such that no submatrix forms an (a, b)-trapping set. This concept was first introduced in the context of low-density parity-check (LDPC) codes in an attempt to estimate the number of redundant rows in a parity-check matrix suitable for iterative decoding. Essentially the same concepts appear in other contexts as ll such as robust syndrome extraction for quantum error correction. Among the known upper bounds on the trapping redundancy, the strongest one was proposed by employing a porful tool in probabilistic combinatorics, called the Lovasz Local Lemma. Unfortunately, the proposed proof invoked this tool in a situation where an assumption made in the lemma does not necessarily hold. Hence, although  do not doubt that nonetheless the proposed bound actually holds, for it to be a mathematical theorem, a more rigorous proof is desired. Another disadvantage of the proposed bound is that it is only applicable to (a, b)-trapping sets with rather small a. Here,  give a more general and sharper upper bound on trapping redundancy by making mathematically more rigorous use of probabilistic combinatorics without relying on the lemma. Our bound is applicable to all potentially avoidable (a, b)-trapping sets with a smaller than the minimum distance of a given linear code, while being generally much sharper than the bound through the Lovasz Local Lemma. In fact, our upper bound is sharp enough to exactly determine the trapping redundancy for many cases, thereby providing precise knowledge in the form of a more general bound with mathematical rigor.
['Yu Tsunoda', 'Yuichiro Fujiwara']
I am writing a research paper on a idea Probabilistic bounds on the trapping redundancy of linear codes
[{"from": "human", "value": "I am writing a research paper on a idea Probabilistic bounds on the trapping redundancy of linear codes"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The trapping redundancy of a linear code is the number of rows of a smallest parity-check matrix such that no submatrix forms an (a, b)-trapping set. This concept was first introduced in the context of low-density parity-check (LDPC) codes in an attempt to estimate the number of redundant rows in a parity-check matrix suitable for iterative decoding. Essentially the same concepts appear in other contexts as ll such as robust syndrome extraction for quantum error correction. Among the known upper bounds on the trapping redundancy, the strongest one was proposed by employing a porful tool in probabilistic combinatorics, called the Lovasz Local Lemma. Unfortunately, the proposed proof invoked this tool in a situation where an assumption made in the lemma does not necessarily hold. Hence, although  do not doubt that nonetheless the proposed bound actually holds, for it to be a mathematical theorem, a more rigorous proof is desired. Another disadvantage of the proposed bound is that it is only applicable to (a, b)-trapping sets with rather small a. Here,  give a more general and sharper upper bound on trapping redundancy by making mathematically more rigorous use of probabilistic combinatorics without relying on the lemma. Our bound is applicable to all potentially avoidable (a, b)-trapping sets with a smaller than the minimum distance of a given linear code, while being generally much sharper than the bound through the Lovasz Local Lemma. In fact, our upper bound is sharp enough to exactly determine the trapping redundancy for many cases, thereby providing precise knowledge in the form of a more general bound with mathematical rigor."}]