14 problems

Belief propagation (BP) is understood through its fixed points, and PCS-style analysis concerns the study of instanton spectra using an accelerated method. BP analysis conjecture.…

Consider the binary Type-II LDPC construction with $l=4$, where $d$ denotes the minimum distance of the code and $d_L$ denotes its minimum pseudo-weight under the relevant decoding…

Let $\mathbf{H}=\mathbf{H}_{\operatorname{PG}(2,q)}$. Consider minimal pseudo-codewords having minimum AWGNC pseudo-weight among all minimal pseudo-codewords that are not multiples…

Let $q$ be an even prime power and let $\mathbf{H}=\mathbf{H}_{\operatorname{PG}(2,q)}$. A component is significant if, for most lines passing through its point, its value is the s…

Consider the construction described in the preceding examples: start with minimal codewords or pseudo-codewords, add them, and modify selected low-valued components to obtain a min…

Let $\mathbf{H}=\mathbf{H}_{\operatorname{PG}(2,q)}$ and let $\boldsymbol{\omega}\in\mathcal{K}(\mathbf{H})$ be a minimal pseudo-codeword of type $\mathbf{t}$, where $t_0$ is non-n…

Let a linear-programming decoding scheme have pseudo-codewords with components $\sigma_i^{(k_*)}$, and define the estimated effective distance by … where the minimum is taken over…

Let $\mathcal{C}$ be a type-I projective-geometry LDPC code, and consider its minimal pseudo-codewords that are not codewords. A minimal pseudo-codeword is called low-support, low-…

Let $\mathcal{C}$ be a type-I projective-geometry LDPC code over a projective plane of order $q$, and let $\mathbf{\boldsymbol{\omega}}$ be a minimal pseudo-codeword that is not a…

Type conjecture. The type $\boldsymbol{t}$ of these pseudo-codewords has $t_0$ non-negative, $t_1$ positive, $t_2$ positive, and $t_\ell=0$ otherwise. If this is false, the smalles…

Structural conjecture. Every minimal pseudo-codeword is a sum of a few minimal pseudo-codewords, with one or two low-value components changed so that they become the large componen…

Minimal pseudo-codeword construction conjecture. For every $q=2^s$, it is possible to construct a minimal pseudo-codeword of type $\boldsymbol{t}$ with $t_1=q+2$,…

Consider a failed LP decoding instance with an identified critical loop, and select a special bit on that loop for a bit-local correction of the LP algorithm. Bit-local correction…

Let $r({\it C})$ denote the loop contribution associated with a loop ${\it C}$ in the loop-series expansion of the partition function, and let low-weight pseudo-codewords correspon…