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I think m_ is the length and n_ is the dimension since m_ > n_.
ExAccCode is a variant of Dual-LPN assumption, as described in the paper Correlated Pseudorandomness from Expand-Accumulate Codes.
To put it simple, Dual-LPN assumption is a problem to distinguish a "compressed vector" $\mathbf u = \mathbf G \cdot \mathbf e$ from the uniformly random vector $\mathbf u'$ drawn from $\lbrace 0,1\rbrace^{n}$. Here, $\mathbf G \in \lbrace 0 , 1 \rbrace^{n \times m}$ and $\mathbf e \in \lbrace 0 , 1 \rbrace^{m}$ is a noise vector.
Based on this, ExAccCode would that accepts an input vector of length m_ and produces an output vector of length n_.
In ea_code.h,
I think m_ is the length and n_ is the dimension since m_>n_.
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