TY - JOUR

T1 - Feedback capacity and coding for the RLL Input-Constrained BEC

AU - Peled, Ori

AU - Sabag, Oron

AU - Permuter, Haim H.

N1 - Funding Information: Manuscript received December 7, 2017; revised December 8, 2018; accepted January 26, 2019. Date of publication March 5, 2019; date of current version June 14, 2019. This work was supported in part by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 337752, in part by the ISF research grant 818/17, and in part by the German Research Foundation (DFG) via the German-Israeli Project Cooperation [DIP]. This paper was presented at the 2017 IEEE International Symposium on Information Theory (ISIT 2017) [1]. Publisher Copyright: © 1963-2012 IEEE.

PY - 2019/3/5

Y1 - 2019/3/5

N2 - The input-constrained binary erasure channel (BEC) with strictly causal feedback is studied. The channel input sequence must satisfy the (0,k) -runlength limited (RLL) constraint, i.e., no more than k consecutive '0's are allowed. The feedback capacity of this channel is derived for all k\geq 1 , and is given by C^{\mathrm {fb}}-{(0,k)}(\varepsilon ) = \max \frac {\overline {\varepsilon }H-{2}(\delta -{0})+\sum -{i=1}^{k-1}\left ({\overline {\varepsilon }^{i+1}H-{2}(\delta -{i})\prod -{m=0}^{i-1}\delta -{m}}\right )}{1+\sum -{i=0}^{k-1}\left ({\overline {\varepsilon }^{i+1} \prod -{m=0}^{i}\delta -{m}}\right )} , where \varepsilon is the erasure probability, \overline {\varepsilon }=1-\varepsilon and H-{2}(\cdot ) is the binary entropy function. The maximization is only over \delta -{k-1} , while the parameters \delta -{i} for i\leq k-2 are straightforward functions of \delta -{k-1}. The lower bound is obtained by constructing a simple coding for all k\geq 1. It is shown that the feedback capacity can be achieved using zero-error, variable length coding. For the converse, an upper bound on the non-causal setting, where the erasure is available to the encoder just prior to the transmission, is derived. This upper bound coincides with the lower bound and concludes the search for both the feedback capacity and the non-causal capacity. As a result, non-causal knowledge of the erasures at the encoder does not increase the feedback capacity for the (0,k) -RLL input-constrained BEC. This property does not hold in general: the (2,\infty ) -RLL input-constrained BEC, where every '1' is followed by at least two '0's, is used to show that the feedback capacity can be strictly smaller than the non-causal capacity.

AB - The input-constrained binary erasure channel (BEC) with strictly causal feedback is studied. The channel input sequence must satisfy the (0,k) -runlength limited (RLL) constraint, i.e., no more than k consecutive '0's are allowed. The feedback capacity of this channel is derived for all k\geq 1 , and is given by C^{\mathrm {fb}}-{(0,k)}(\varepsilon ) = \max \frac {\overline {\varepsilon }H-{2}(\delta -{0})+\sum -{i=1}^{k-1}\left ({\overline {\varepsilon }^{i+1}H-{2}(\delta -{i})\prod -{m=0}^{i-1}\delta -{m}}\right )}{1+\sum -{i=0}^{k-1}\left ({\overline {\varepsilon }^{i+1} \prod -{m=0}^{i}\delta -{m}}\right )} , where \varepsilon is the erasure probability, \overline {\varepsilon }=1-\varepsilon and H-{2}(\cdot ) is the binary entropy function. The maximization is only over \delta -{k-1} , while the parameters \delta -{i} for i\leq k-2 are straightforward functions of \delta -{k-1}. The lower bound is obtained by constructing a simple coding for all k\geq 1. It is shown that the feedback capacity can be achieved using zero-error, variable length coding. For the converse, an upper bound on the non-causal setting, where the erasure is available to the encoder just prior to the transmission, is derived. This upper bound coincides with the lower bound and concludes the search for both the feedback capacity and the non-causal capacity. As a result, non-causal knowledge of the erasures at the encoder does not increase the feedback capacity for the (0,k) -RLL input-constrained BEC. This property does not hold in general: the (2,\infty ) -RLL input-constrained BEC, where every '1' is followed by at least two '0's, is used to show that the feedback capacity can be strictly smaller than the non-causal capacity.

KW - Constrained coding

KW - feedback capacity

KW - finite-state machine

KW - markov decision process

KW - posterior matching

KW - runlength limited (RLL) constraints

UR - http://www.scopus.com/inward/record.url?scp=85067619255&partnerID=8YFLogxK

U2 - https://doi.org/10.1109/TIT.2019.2903252

DO - https://doi.org/10.1109/TIT.2019.2903252

M3 - Article

SN - 0018-9448

VL - 65

SP - 4097

EP - 4114

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 7

M1 - 8660646

ER -