--- title: Z-Channel Embedding url: https://www.emergentmind.com/topics/z-channel-embedding type: topic --- # Z-Channel Embedding Searching arXiv for the cited Z-channel papers to ground the article in the primary literature. Z-channel embedding denotes the design of encoding and decoding schemes that reliably store or transmit information over a binary Z-channel, an asymmetric channel in which only \(1\to 0\) flips are permitted and \(0\to 1\) flips are prohibited. In the adversarial formulation developed in recent work, a length-\(n\) binary codeword is subjected to at most \(\tau n\) asymmetric errors, and the central problem is to determine how much information can be embedded with zero decoding error under worst-case corruption [2010.16362, 2105.01427]. Within this setting, the literature distinguishes non-adaptive codes, list-decodable codes, and two-stage strategies with one noiseless feedback use. These works establish threshold phenomena analogous to Plotkin-type points, derive combinatorial bounds on code size above and below those thresholds, and exhibit explicit two-stage constructions that use free-point allocations or list-resolution mechanisms to exploit the channel’s asymmetry [2010.16362, 2105.01427, 2202.00136]. ## 1. Z-channel model and geometric framework The Z-channel is binary and asymmetric: a transmitted \(0\) is always received as \(0\), whereas a transmitted \(1\) may be received as either \(1\) or \(0\). In the adversarial model, the encoder sends \(x=(x_1,\ldots,x_n)\in\{0,1\}^n\), and the receiver observes \(y=(y_1,\ldots,y_n)\) subject to $$ y_i \in \begin{cases} \{0\}, & \text{if } x_i = 0,\\ \{0,1\}, & \text{if } x_i = 1. \end{cases} $$ The total number of asymmetric errors is bounded by \(t\le \tau n\), with \(0<\tau<1\) [2010.16362]. Equivalent formulations in the literature describe the adversary as zeroing out up to \(n\tau\) ones and never creating new ones [2105.01427]. Several geometric notions organize code design for this channel. The quantity $$ \Delta(x,y)=|\{i:x_i=1,\ y_i=0\}| $$ counts downward flips from \(x\) to \(y\). One asymmetric distance used in the two-stage analysis is $$ d_Z(x,y):=2\max(\Delta(x,y),\Delta(y,x)), $$ with the relation $$ d_Z(x,y)=d_H(x,y)+|wt(x)-wt(y)|. $$ Hence, on constant-weight codes, \(d_Z\) coincides with the Hamming distance [2010.16362]. A related formulation writes $$ d_Z(x,y)=\max\{\Delta(x,y),\Delta(y,x)\}, $$ together with the Z-ball $$ B_t^{\mathrm Z}(y)=\{x\in\{0,1\}^n:\Delta(x,y)\le t,\ \Delta(y,x)=0\}, $$ that is, all words obtainable from \(y\) by changing at most \(t\) ones to zeros and never changing zeros to ones [2105.01427]. For list decoding, the center of an \(L\)-tuple has a particularly simple structure. For codewords \(x_1,\ldots,x_L\), the minimizer of the Chebyshev radius is the bitwise AND, equivalently the vector supported on the intersection of supports. In constant-weight codes of weight \(nw\), this gives $$ rad(L)=nw-\Bigl|\bigcap_{i\in[L]}\mathrm{supp}(x_i)\Bigr|, $$ which makes support intersections the basic combinatorial object controlling Z-channel ambiguity [2105.01427]. This asymmetry-driven geometry is also what makes weight distributions and support inclusion tests central in practical decoding. ## 2. Decoding regimes: unique decoding, list decoding, and two-stage feedback A non-adaptive asymmetric \((n,M,t)_Z\) code is a subset \(C\subset\{0,1\}^n\) with \(|C|=M\) such that every pair of distinct codewords has Z-distance at least \(2t\); such codes correct up to \(t\) asymmetric errors in the worst-case sense [2202.00136]. In the list-decoding formulation, a code is \((L-1)\)-list-decodable with radius \(\tau\) if every Z-ball of radius \(\lceil \tau n\rceil\) contains at most \(L-1\) codewords [2105.01427]. The two notions differ operationally: unique decoding demands a singleton candidate set, while list decoding permits bounded ambiguity that can later be resolved by auxiliary information. Two-stage embedding introduces a single noiseless feedback use at a designated time \(n_1\), with \(1n_1\), it may transmit \(x_i=x_i(m,y^{n_1})\), where \(y^{n_1}=(y_1,\ldots,y_{n_1})\) is the observed partial output [2010.16362]. Formally, one may write the encoding rule as \[ c(m,y^{n_1})=(c_1(m),c_2(m,y^{n_1})), \] with \(n=n_1+n_2\), and require the decoder to recover the message for all patterns with at most \(t\) asymmetric errors [2202.00136]. The main asymptotic quantity for two-stage coding is $$ R_Z^{(2)}(\tau):=\limsup_{n\to\infty}\frac{\log M_Z^{(2)}(n,\tau n)}{n}, $$ where \(M_Z^{(2)}(n,t)\) is the maximum number of messages achievable with blocklength \(n\), at most \(t\) asymmetric errors, and one feedback moment [2010.16362]. Positive-rate embedding means \(R_Z^{(2)}(\tau)>0\), while zero-rate means \(R_Z^{(2)}(\tau)=0\). A distinct small-\(n\) two-stage framework for the single-error case \(t=1\) uses second-stage codebooks indexed by the first-stage observation and allocates free points to possible parent vertices in a directed graph of one-error transitions [2202.00136]. This construction is combinatorial rather than asymptotic, but it addresses the same operational problem: how to exploit one feedback event to separate “no error in stage 1” from “one stage-1 error occurred” and recover the embedded payload. ## 3. Plotkin-type thresholds and phase transitions For Z-channel list decoding, the literature identifies a threshold analogous to the Plotkin point. For fixed list parameter \(L\ge 2\), define \[ \tau_L(w)=w-w^L,\qquad \tau_{\mathrm{Plotkin}}(L)=\max_{0\le w\le 1}\tau_L(w). \] The maximizer is \[ w_{\max}=L^{-\frac{1}{L-1}}, \] and the resulting threshold is \[ \tau_{\mathrm{Plotkin}}(L)=L^{-\frac{1}{L-1}}-L^{-\frac{L}{L-1}} =(L-1)L^{-\frac{L}{L-1}}. \] For unique decoding, corresponding to \(L=2\), this yields \(w_{\max}=1/2\) and \(\tau_{\mathrm{Plotkin}}(2)=1/4\) [2105.01427]. Below this threshold, the largest \((L-1)\)-list-decodable Z-channel codes have exponential size in \(n\); above it, any such code has size bounded independently of \(n\) [2105.01427]. The two-stage problem has a distinct threshold because the encoder is allowed one adaptive update after feedback. The sharp transition established for two-stage encoding is $$ \tau^\star=\max_{00\ \text{for all}\ \tau<\tau^\star,\qquad R_Z^{(2)}(\tau)=0\ \text{for all}\ \tau>\tau^\star $$ [2010.16362]. Thus, allowing one feedback moment shifts the positive-rate boundary from the unique-decoding Plotkin point \(1/4\) to an asymptotic two-stage threshold near \(0.44\). The converse mechanism combines a list-decoding upper bound for first-stage constant-weight codes with bounds on the best high-error, low-rate codes available for second-stage disambiguation [2010.16362]. A plausible implication is that two-stage feedback fundamentally changes the embedding threshold, but does not eliminate threshold behavior itself: beyond \(\tau^\star\), first-stage ambiguity and second-stage error-correction limits together force zero asymptotic rate. ## 4. Two-stage embedding mechanisms The asymptotic two-stage strategy of "Two-stage coding over the Z-channel" splits the block into \(n_1=\alpha n\) and \(n_2=(1-\alpha)n\), fixes a constant weight \(w\in(0,1)\), and uses a \(w\)-constant-weight code \(C\subset\{0,1\}^{n_1}\) of size \(M=2^{R_1n_1}\) in stage 1 [2010.16362]. The code is required to be \((t,L)_Z\)-list-decodable for a range of list sizes up to \(L_{\text{up}}\). Because only \(1\to 0\) flips occur, if the transmitted stage-1 word has weight \(w_1n_1\), then after receiving \(y_1\) the receiver can determine the exact number of stage-1 errors from \[ wt(y_1)=w_1n_1-\tau_1n_1. \] Hence both parties know the remaining error budget \(\tau_2n_2=\tau n-\tau_1n_1\) [2010.16362]. When \(\tau_1\) is small enough that the Z-ball around the stage-1 output intersects at most \(L\) codewords, the second stage uses a high-error, low-rate code of size \(L\) to resolve the candidate list. Decoding succeeds if $$ \tau_2\le \tau_Z(L)-\varepsilon, $$ where \(\tau_Z(M)\) is the supremum fraction of correctable asymmetric errors for codes of size \(M\) [2010.16362]. For example, exact small-\(M\) values include \(\tau_Z(3)=1/2\), \(\tau_Z(4)=1/2\), \(\tau_Z(5)=2/5\), \(\tau_Z(6)=2/5\), \(\tau_Z(7)=3/8\), \(\tau_Z(8)=4/11\), and \(\tau_Z(12)=1/3\) [2010.16362]. When \(\tau_1\) is larger and the first-stage list may be exponentially large, the construction shortens the stage-1 code by fixing the positions of the ones observed in \(y_1\), derives an upper bound on the logarithmic size exponent of the shortened code, and then chooses a second-stage random constant-weight code of normalized weight \(1/2\) with rate $$ R_2(\alpha,\tau_1,w,w_1,R_1,\varepsilon) := \frac{\alpha (1-w_1+\tau_1)}{1-\alpha}\left[ h\!\left(\frac{\tau_1}{1-w_1+\tau_1}\right) - h\!\left(\frac{1-\sqrt{\,1 - \frac{4(\tau^*(R_1,1,w)-\varepsilon)}{1+w_1-\tau_1}\,}{2}\right) \right]. \tag{2} $$ Decoding succeeds if $$ \tau_2\le \tau^*(R_2,1,1/2)-\varepsilon $$ [2010.16362]. In operational terms, stage 2 either transmits a finite disambiguation index when the first-stage list is small, or transmits fresh positive-rate information while simultaneously resolving residual ambiguity when the list is large. The small-block construction of "Non-adaptive and two-stage coding over the Z-channel" uses a different two-stage mechanism specialized to \(t=1\). For each first-stage observation \(v\), a second-stage code \(C(v)\) of length \(n_2\) has \(M(v)\) codewords and \(F(v)\) free points. Free points are strings not covered by any radius-1 ball around a codeword. The decoder checks whether the received second-stage word lies in \(\bigcup_{a\in C(v)}B_1(a)\). If it does, it declares “no stage-1 error” and decodes to a center; otherwise, it declares “stage-1 error occurred” and uses a map from free points to parent vertices [2202.00136]. This realizes the same separation between error patterns and payload resolution, but with explicit finite-block codebooks. ## 5. Bounds on code size and rate A central result for codes above the list-decoding Plotkin point is the sharp scaling law \[ |\mathcal C|=\Theta_L(\epsilon^{-3/2}) \qquad\text{for}\qquad \tau=\tau_{\mathrm{Plotkin}}(L)+\epsilon, \] valid for sufficiently small \(\epsilon>0\) and all \(L\ge 2\) [2105.01427]. Thus, above the threshold, code size becomes independent of blocklength and is controlled by the distance \(\epsilon\) above the threshold. For unique decoding, this yields the same \(\Theta(\epsilon^{-3/2})\) behavior above \(1/4\), and a concrete lower bound is \[ |\mathcal C|\ge \frac{3\sqrt{3}}{128}\,\epsilon^{-3/2}(1+o(1)) \quad\text{as }\epsilon\to 0 \] [2105.01427]. For constant-weight codes above the threshold, a double-counting argument gives an upper bound of order \(O_L(\epsilon^{-1})\) per weight slice. The global \(\Theta_L(\epsilon^{-3/2})\) law arises by a non-uniform stratification into \(O_L(\epsilon^{-1/2})\) weight slices near and away from the maximizing weight \(w_{\max}\) [2105.01427]. This suggests that the dominant combinatorial difficulty above the threshold is not confined to a single weight level, but to a neighborhood of the critical weight where the constraint is tightest. Below the Plotkin point, the maximum code size is exponential in \(n\), and the literature provides rate bounds for constant-weight codes. One upper bound has the form \[ R(C)\le \min_{v,a} I(w,v,a)+o(1), \] subject to \[ (1-v)\left[\frac{w-a}{1-v}-\left(\frac{w-a}{1-v}\right)^L\right] +v\left[\frac{a}{v}-\left(\frac{a}{v}\right)^L\right]\le \tau, \] where \(I(w,v,a)\) is the mutual information of a binary pair \((U,X)\) with joint distribution \[ P_{U,X}= \begin{bmatrix} 1-w-v+a & v-a\\ w-a & a \end{bmatrix} \] [2105.01427]. A lower bound from random coding with expurgation is \[ R\ge \frac{1}{L-1}\min_{P\in\mathcal K(w,\tau)} D\!\left(P\big\|\mathrm{Ber}(w)^{\otimes L}\right), \] where \(\mathcal K(w,\tau)\) is the confusability set of joint distributions whose \(L\)-fold all-one mass is at least \(w-\tau\) [2105.01427]. For the two-stage problem, the first-stage list-decoding component is governed by the normalized \(L\)-radius \(\tau^*(R,L,w)\), defined through explicit functions \(g(h_L,L,w)\) and \(\delta(h_L,L,w)\). The random coding theorem states that for any fixed \(w\in(0,1)\), \(L\le L_{\text{up}}\), and \(Rw-w^{L+1}\), \[ \frac{M^L}{(M-1)\cdots(M-L)}\ge \frac{\tau}{w-w^{L+1}}, \] which forces vanishing rate beyond \(w-w^{L+1}+\varepsilon\) for every fixed \(\varepsilon>0\) [2010.16362]. ## 6. High-error low-rate codes and explicit finite-block constructions The finite-size quantity \[ \tau_Z(M):=\sup\{\tau:\exists\ \text{code of size }M\text{ correcting a fraction }\tau\text{ of asymmetric errors}\} \] plays two roles: it bounds the second stage of asymptotic two-stage schemes, and it describes the best achievable correction level when codebook size is fixed [2010.16362]. For \(M\ge 2\), \(\tau_Z(M)\) has an LP characterization in terms of a binary matrix \(D\) indexed by unordered pairs from \([M]\) and \(2^M\) columns: \[ \tau_Z(M)^{-1}=\max_{(*)}\sum_{i=1}^{\binom{M}{2}} y_i, \] where the maximum is over nonnegative vectors \(y\) such that each entry of \(yD\) is at most \(1\) [2010.16362]. The same paper proves that for any \(\varepsilon>0\) and \(M\ge 1\), there exists \(n_0\) such that for all \(n>n_0\) there is a length-\(n\) code of size \(M\) correcting a fraction \(\tau_Z(M)-\varepsilon\) of asymmetric errors. A key side result is a Plotkin-type upper bound for asymmetric error-correcting codes: for \(\varepsilon>0\), any binary code correcting a fraction \(1/4+\varepsilon\) of asymmetric errors contains at most \(O(\varepsilon^{-3/2})\) codewords, and \(\tau_Z(M)\to 1/4\) as \(M\to\infty\) [2010.16362]. This complements the above-threshold scaling law for list decoding and shows that \(1/4\) remains the asymptotic unique-decoding barrier for fixed-size high-error codes. The finite-block paper supplies explicit two-stage constructions for \(t=1\). One example uses \(n_1=5\), \(n_2=4\), and weight-dependent second-stage codes: - for \(w\in\{0,1\}\), a \((4,2,12,1)_Z\) codebook \(\{0000,0011\}\); - for \(w\in\{2,3\}\), a \((4,3,9,1)_Z\) codebook \(\{0000,0011,1100\}\); - for \(w\in\{4,5\}\), a \((4,4,4,1)_Z\) codebook \(\{0000,0011,1100,1111\}\). The weight-class feasibility inequalities \[ (5-0)\cdot 2\le 12,\quad 4\cdot 3\le 12,\quad 3\cdot 3\le 9,\quad 2\cdot 4\le 9,\quad 1\cdot 4\le 4 \] hold, so the total message count is \[ M_{\text{total}}= \binom50\cdot2+\binom51\cdot2+\binom52\cdot3+\binom53\cdot3+\binom54\cdot4+\binom55\cdot4=96, \] with rate \[ R=\frac{\log_2 96}{9}\approx 0.732 \] [2202.00136]. Another example with \(n=8\) achieves \(M_{\text{total}}=53\) and \(R\approx 0.716\) [2202.00136]. For \(t=1\), the same paper reports that two-stage strategies exceed the cited full-feedback adaptive strategies for some small lengths: - \(n=5\): two-stage \(M=9\) versus full-feedback \(M=8\); - \(n=8\): two-stage \(M=53\) versus full-feedback \(M=32\); - \(n=9\): two-stage \(M=96\) versus full-feedback \(M=64\); - \(n=10\): two-stage \(M=177\) versus full-feedback \(M=128\) [2202.00136]. This is a finite-block phenomenon; it should not be conflated with the asymptotic two-stage threshold near \(0.44\). ## 7. Relation to probabilistic models, implementation guidance, and open issues The cited works are primarily combinatorial and adversarial. They differ from the probabilistic Z-channel in which each transmitted \(1\) is flipped to \(0\) independently with probability \(p\). In that stochastic model, with Bernoulli-\(q\) input, \[ I(X;Y)=h_2(q(1-p))-q\,h_2(p), \] and the capacity is \[ C_Z(p)=\max_{0\le q\le 1}\bigl[h_2(q(1-p))-q\,h_2(p)\bigr] \] [2202.00136]. Another equivalent capacity expression quoted in the literature is \[ C(\tau)=\log\!\Bigl(1+\tau^{\frac{\tau}{1-\tau}}(1-\tau)\Bigr) \] up to equivalent forms from prior work [2105.01427]. These stochastic capacities are not the same object as adversarial list-decoding capacity; the latter depends on worst-case support intersections and does not generally coincide with the Shannon model [2105.01427]. For practical embedding over an adversarial Z-channel, the literature repeatedly emphasizes constant-weight organization and support-based decoding. In constant-weight blocks, a candidate codeword \(x\) is compatible with a received word \(y\) precisely when \(\mathrm{supp}(y)\subseteq \mathrm{supp}(x)\) and the weight loss is within the error budget [2105.01427]. In two-stage schemes, feedback reveals the first-stage error count through the received weight, and this determines whether the second stage should perform bounded-list disambiguation or use a positive-rate fallback code [2010.16362]. The small-\(n\) framework refines this into a balls-versus-free-points test on the second-stage suffix [2202.00136]. Several limitations remain explicit in the current literature. Exact code size at the threshold \(\tau=\tau_{\mathrm{Plotkin}}(L)\) is open; a conjectured behavior is \(\Theta_L(n^{3/2})\) [2105.01427]. Tight constants in the \(\Theta_L(\epsilon^{-3/2})\) regime above the threshold are not optimized [2105.01427]. For larger \(n\) and \(t>1\), the LP-based free-point search of the small-block two-stage method becomes more complex [2202.00136]. The asymmetry of Z-distance also prevents a direct application of Delsarte-type LP methods based on shift-invariant sphere intersections, leaving alternative LP-like tools as an open direction [2105.01427]. Taken together, these results define Z-channel embedding as a coding-theoretic program centered on asymmetric support geometry, threshold behavior, and adaptive disambiguation. The non-adaptive regime is controlled by the Z-channel Plotkin point, the above-threshold regime exhibits the sharp \(\Theta_L(\epsilon^{-3/2})\) law, and one-use feedback raises the positive-rate boundary to \[ \tau^\star=\max_{0