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The bin index of the deterministic output is selected by the transmitter, such that the relay's transmission is coordinated with the states.
This coding scheme also applies for the MAC with partial cribbing and non-causal CSI at one transmitter and receiver.
Partial MDS (PMDS) codes are a class of erasure-correcting array codes which combine local correction of the rows with global correction of the array.
An $m\times n$ array code is called an $(r;s)$ PMDS code if each row belongs to an $[n,n-r,r 1]$ MDS code and the code can correct erasure patterns consisting of $r$ erasures in each row together with $s$ more erasures anywhere in the array.
The transmission is split to blocks; in each block, the relay decodes a part of the message and cooperation is established using those bits.
When the channel depends on a state, the decoding procedure at the relay reduces the transmission rate.
This paper considers the problem of designing maximum distance separable (MDS) codes over small fields with constraints on the support of their generator matrices.
For any given $m\times n$ binary matrix $M$, the , states that if $M$ satisfies the so-called MDS condition, then for any field $\mathbb$ of size $q\geq n m-1$, there exists an $[n,m]_q$ MDS code whose generator matrix $G$, with entries in $\mathbb$, fits $M$ (i.e., $M$ is the support matrix of $G$).
The CSI is available only at the transmitter and receiver, but not at the relay.
We show that for each of the original classes of TCs, it is possible to find an equivalent ensemble by proper selection of the design parameters in the unified ensemble.
We also derive the density evolution (DE) equations for this ensemble over the binary erasure channel.
We demonstrate the strength of our technique by proving the TM-MDS conjecture for the cases where the number of rows ($m$) of $M$ is upper bounded by $.
For this class of special cases of $M$ where the only additional constraint is on $m$, only cases with $m\leq 4$ were previously proven theoretically, and the previously used proof techniques are not applicable to cases with $m In this paper we propose a woven block code construction based on two convolutional outer codes and a single inner code.