From 56e3a0e5ca93494792e14f37b70b99ef37f0201d Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Mon, 4 May 2026 20:21:21 +0200 Subject: [PATCH] Consistently capitalize character after semicolon --- src/thesis/chapters/2_fundamentals.tex | 2 +- src/thesis/chapters/4_decoding_under_dems.tex | 40 +++++++++---------- .../chapters/5_conclusion_and_outlook.tex | 10 ++--- src/thesis/chapters/abstract.tex | 2 +- 4 files changed, 27 insertions(+), 27 deletions(-) diff --git a/src/thesis/chapters/2_fundamentals.tex b/src/thesis/chapters/2_fundamentals.tex index 5a18351..be5ce44 100644 --- a/src/thesis/chapters/2_fundamentals.tex +++ b/src/thesis/chapters/2_fundamentals.tex @@ -193,7 +193,7 @@ decoding \cite[Preface]{ryan_channel_2009}. The iterative decoding algorithms are generally defined in terms of message passing on the \textit{Tanner graph} of a code. The Tanner graph is a bipartite graph that constitutes an alternative representation of the \ac{pcm}. -We define two types of nodes: \acp{vn}, corresponding to codeword +We define two types of nodes: \Acp{vn}, corresponding to codeword bits, and \acp{cn}, corresponding to individual parity checks. We then construct the Tanner graph by connecting each \ac{cn} to the \acp{vn} that make up the corresponding parity check diff --git a/src/thesis/chapters/4_decoding_under_dems.tex b/src/thesis/chapters/4_decoding_under_dems.tex index ec1f568..0a9bc07 100644 --- a/src/thesis/chapters/4_decoding_under_dems.tex +++ b/src/thesis/chapters/4_decoding_under_dems.tex @@ -470,7 +470,7 @@ model and is difficult to predict beforehand. The block-diagonal structure reflects the time-like locality of the syndrome extraction circuit, with each block corresponding to one syndrome measurement round. - Two consecutive windows are highlighted: the window size $W + Two consecutive windows are highlighted: The window size $W \in \mathbb{N}$ controls the number of syndrome rounds included in each window, while the step size $F \in \mathbb{N}$ controls how many rounds separate the start of @@ -1013,8 +1013,8 @@ the most reliable \ac{vn}, meaning we perform a hard decision and remove it from the following decoding process. This means that when moving from one window to the next, we now have -more information available: not just the \ac{bp} messages but also the -information about what \acp{vn} were decimated and to what values. +more information available: Not just the \ac{bp} messages but also the +Information about what \acp{vn} were decimated and to what values. We call this \emph{decimation information} in the following. We can extend \Cref{alg:warm_start_bp} by additionally passing the decimation information after initializing the \ac{cn} to \ac{vn} messages. @@ -1404,7 +1404,7 @@ The fact that the $W = 5$ curve is already very close to the whole-block decoder indicates that the marginal benefit of enlarging the window saturates after a certain point. Thus, from a practical standpoint, the choice of $W$ represents a -trade-off between decoding latency and accuracy: larger windows +trade-off between decoding latency and accuracy: Larger windows delay the start of decoding by requiring more syndrome extraction rounds to be collected upfront, while the diminishing returns above $W = 4$ suggest that growing the window much further yields little @@ -1511,7 +1511,7 @@ The dashed colored curves reproduce the cold-start results from corresponding warm-start runs for the same window sizes $W \in \{3, 4, 5\}$. The remaining experimental parameters are unchanged: -the step size is fixed to $F = 1$, +The step size is fixed to $F = 1$, the inner \ac{bp} decoder is allowed up to $200$ iterations per window invocation, the black curve again gives the whole-block reference, and the physical error rate is swept from $p = 0.001$ to @@ -1707,7 +1707,7 @@ $n_\text{iter} \in [32, 512]$. All curves decrease monotonically with the iteration budget, but contrary to our expectation, none of them appears to fully saturate -within the swept range: even at $n_\text{iter} = 4096$, every curve +within the swept range: Even at $n_\text{iter} = 4096$, every curve still exhibits a noticeable downward slope. At $n_\text{iter} = 32$, the whole-block curve lies below both the $W=4$ and $W=5$ sliding-window curves. @@ -1729,7 +1729,7 @@ mirroring the behavior already observed in \Cref{fig:whole_vs_cold_vs_warm}. These observations are largely consistent with the effective-iterations hypothesis put forward above. The whole-block decoder eventually overtaking every windowed scheme -matches the prediction made there: with a sufficiently large +matches the prediction made there: With a sufficiently large iteration budget, the whole-block decoder reaches an error rate that none of the windowed schemes can beat, because of the more global nature of the considered constraints. @@ -1767,7 +1767,7 @@ sliding-window approach is still at an advantage. Having examined the effect of the window size $W$, we next turn to the second windowing parameter, the step size $F$. We carry out an investigation analogous to the one above: -we first compare warm- and cold-start decoding across the full range +We first compare warm- and cold-start decoding across the full range of physical error rates at a fixed iteration budget, and then we examine the dependence on the iteration budget at a fixed physical error rate. @@ -1994,7 +1994,7 @@ At fixed $F$, the warm-start approach lies below cold-start across the entire sweep, and at fixed warm or cold start, smaller $F$ produces a lower \ac{ler}. Both gaps grow as the physical error rate decreases: -the curves at $F = 1$ separate further from those at $F = 2$ and $F = 3$, +The curves at $F = 1$ separate further from those at $F = 2$ and $F = 3$, and the warm-start curves separate further from the cold-start ones. In \Cref{fig:bp_f_over_iter}, all six curves again decrease monotonically with the iteration budget, with no clear saturation @@ -2016,7 +2016,7 @@ With $W$ held fixed, decreasing $F$ enlarges the overlap between consecutive windows from $W - F$ to $W - F + 1$ syndrome measurement rounds, so a smaller step size is beneficial for the same reason that a larger window size is: -each \ac{vn} in an overlap region participates in more window +Each \ac{vn} in an overlap region participates in more window invocations, and the warm-start modification effectively accumulates iterations on it across these invocations. The widening of the warm/cold gap towards low iteration counts and @@ -2281,7 +2281,7 @@ This is the opposite of what we observed for plain \ac{bp}, where warm-start improved upon cold-start at every parameter setting. The gap between the warm- and cold-start curves additionally widens as the physical error rate decreases: -at the lowest sampled rate $p = 0.001$, the per-round \ac{ler} of the +At the lowest sampled rate $p = 0.001$, the per-round \ac{ler} of the warm-start runs is more than two orders of magnitude above that of the corresponding cold-start runs. In \Cref{fig:bpgd_w}, larger window sizes yield lower per-round @@ -2300,13 +2300,13 @@ than its cold-start counterpart is surprising in light of the results for plain \ac{bp}, where the warm-start modification was uniformly beneficial. The dependence on the window size in \Cref{fig:bpgd_w} is, on its own, consistent with the same explanation that we gave for -\Cref{fig:whole_vs_cold}: larger windows expose the inner decoder to +\Cref{fig:whole_vs_cold}: Larger windows expose the inner decoder to a larger fraction of the constraints encoded in the detector error matrix at the time of decoding, and this benefits both warm- and cold-start decoding. The dependence on the step size in \Cref{fig:bpgd_f}, however, is the opposite of the corresponding dependence under plain \ac{bp} -(\Cref{fig:bp_f_over_p}): for warm-start, smaller $F$ now degrades performance +(\Cref{fig:bp_f_over_p}): For warm-start, smaller $F$ now degrades performance rather than helps, even though smaller $F$ implies a larger overlap in both cases. @@ -2564,7 +2564,7 @@ the warm-start curves now show a clear reordering as $n_\text{iter}$ grows. At low iteration budgets the warm-start ordering matches the cold-start ordering, with $F = 1$ best and $F = 3$ worst, but at the -largest iteration budget this ordering is fully inverted: warm-start +largest iteration budget this ordering is fully inverted: Warm-start $F = 1$ is now the worst and $F = 3$ the best. % [Interpretation] Figure 4.11 @@ -2596,7 +2596,7 @@ decoding performance. The same mechanism explains the inversion of the step-size ordering in \Cref{fig:bpgd_iter_F}. At low iteration budgets, the ordering is set by the same overlap -argument as for plain \ac{bp}: smaller $F$ implies a larger overlap +argument as for plain \ac{bp}: Smaller $F$ implies a larger overlap between consecutive windows, more shared messages, and therefore better warm-start performance. At large iteration budgets, the ordering is set by the premature hard @@ -2777,7 +2777,7 @@ since the decimation decisions were made based on the messages themselves. \Cref{fig:bpgd_msg} repeats the experiment of \Cref{fig:bpgd_wf} with the modified warm-start procedure that carries over only the \ac{bp} messages. -All other experimental parameters are unchanged: the maximum number +All other experimental parameters are unchanged: The maximum number of inner \ac{bp} iterations is $n_\text{iter} = 5000$, and the physical error rate is swept from $p = 0.001$ to $p = 0.004$ in steps of $0.0005$. @@ -2810,7 +2810,7 @@ the warm-start regression observed in \Cref{fig:bpgd_wf}, and warm-start now consistently outperforms cold-start. The dependence on the window size and the step size also recovers the qualitative behavior we observed for plain \ac{bp} in -\Cref{fig:whole_vs_cold_vs_warm,fig:bp_f_over_p}: a larger overlap +\Cref{fig:whole_vs_cold_vs_warm,fig:bp_f_over_p}: A larger overlap between consecutive windows, achieved either by enlarging $W$ or by decreasing $F$, both improves the absolute decoding performance and increases the warm-start advantage over cold-start. @@ -2994,7 +2994,7 @@ cold-start curves across the entire range of $n_\text{iter}$ available to us. \Cref{fig:bpgd_msg_iter} repeats the experiment of \Cref{fig:bpgd_iter} with the modified warm-start procedure that carries over only the \ac{bp} messages. -All other experimental parameters are unchanged: the physical error +All other experimental parameters are unchanged: The physical error rate is fixed at $p = 0.0025$ and the iteration budget is swept over $n_\text{iter} \in \{32, 128, 256, 512, 1024, 1536, 2048, 2560, 3072, 3584, 4096\}$. @@ -3026,7 +3026,7 @@ initialization no longer freezes any \acp{vn} in the next window. The dependence of this benefit on $W$ and $F$ also recovers the pattern observed for plain \ac{bp} in \Cref{fig:whole_vs_cold_vs_warm,fig:bp_f_over_p}: -larger overlap, achieved by larger $W$ or smaller $F$, yields more +Larger overlap, achieved by larger $W$ or smaller $F$, yields more effective extra iterations and therefore a larger warm-start gain. % BPGD conclusion @@ -3048,7 +3048,7 @@ cold-start that follows the same behavior as for plain \ac{bp} with regard to overlap. A second observation specific to \ac{bpgd} is that its iteration requirements are substantially larger than those of plain \ac{bp}: -the per-round \ac{ler} drops sharply only once the iteration budget +The per-round \ac{ler} drops sharply only once the iteration budget is on the order of the number of \acp{vn} in each window. Future work could include a softer treatment of the decimation state diff --git a/src/thesis/chapters/5_conclusion_and_outlook.tex b/src/thesis/chapters/5_conclusion_and_outlook.tex index fdc9408..5cf1b5b 100644 --- a/src/thesis/chapters/5_conclusion_and_outlook.tex +++ b/src/thesis/chapters/5_conclusion_and_outlook.tex @@ -7,7 +7,7 @@ This thesis investigates decoding under \acp{dem} for fault-tolerant \ac{qec}, with a focus on low-latency decoding methods for \ac{qldpc} codes. The repetition of the syndrome measurements, especially under consideration of circuit-level noise, leads to a significant increase -in decoding complexity: in our experiments on the $\llbracket +in decoding complexity: In our experiments on the $\llbracket 144,12,12 \rrbracket$ \ac{bb} code with $12$ syndrome extraction rounds, the check matrix grows from 144 \acp{vn} and 72 \acp{cn} to 9504 \acp{vn} and 1008 \acp{cn}. @@ -46,18 +46,18 @@ min-sum algorithm. For standard min-sum \ac{bp}, the warm start is consistently beneficial to the cold start, across the considered parameter ranges. The size of the gain depends on the overlap between consecutive -windows: enlarging $W$ or shrinking $F$, both of which enlarge the +windows: Enlarging $W$ or shrinking $F$, both of which enlarge the overlap, result in larger gains of the warm-start. We observe that the underlying mechanism is an effective increase in the number of \ac{bp} iterations spent on the \acp{vn} in the overlap -region: each such \ac{vn} is processed by multiple consecutive window +region: Each such \ac{vn} is processed by multiple consecutive window invocations, and the warm start lets these invocations accumulate iterations on the same \acp{vn} rather than restarting from scratch. The gain was most pronounced at low numbers of maximum iterations, where every additional iteration carries proportionally more information. For \ac{bpgd}, we note that more information is available in the -overlap region of a window: in addition to the \ac{bp} messages, +overlap region of a window: In addition to the \ac{bp} messages, there is information about which \acp{vn} were decimated and to what value. Passing this decimation information to the next window in addition to the messages turned out to worsen the performance considerably, which @@ -66,7 +66,7 @@ overlap region. Restricting the warm start to the \ac{bp} messages alone, removed this effect. The resulting message-only warm start recovered a consistent improvement over cold-start that followed the same qualitative -behaviour as for standard \ac{bp}: larger overlap, achieved by larger +behaviour as for standard \ac{bp}: Larger overlap, achieved by larger $W$ or smaller $F$, yielded a larger gain, and the performance difference is most pronounced at low numbers of maximum iterations. diff --git a/src/thesis/chapters/abstract.tex b/src/thesis/chapters/abstract.tex index 61df5cc..6b824f4 100644 --- a/src/thesis/chapters/abstract.tex +++ b/src/thesis/chapters/abstract.tex @@ -49,7 +49,7 @@ For both standard \ac{bp} and \ac{bpgd} decoding, the warm-start initialization provides a consistent improvement across all examined parameter settings. We attribute this to an effective increase in \ac{bp} iterations on -variable nodes in the overlap regions: each such VN is processed by +variable nodes in the overlap regions: Each such VN is processed by multiple consecutive windows, and warm-starting lets these invocations accumulate iterations rather than restart from scratch. Crucially, the warm-start modification incurs no additional