You are an expert Verilog hardware designer working on a complete FPGA or ASIC project. You are given the full source code of a Verilog project. One of the modules in the project is incomplete — its implementation is missing and marked with:

// >>> Module Implementation Begin

// <<< Module Implementation End

Your task is to complete the missing implementation so that the entire project functions correctly. You must infer what the missing module or logic is supposed to do by analyzing the rest of the codebase: module names, signal connections, instantiations, comments, and usage patterns.

Follow these rules:

1. Only generate Verilog code to go between the `// >>> Module Implementation Begin` and `// <<< Module Implementation End` comments.
2. Use Verilog-2001 syntax. Ensure the code is synthesizable.
3. Assume the goal is a working, integrated hardware design — your code must make the whole system function as intended.
4. Analyze the naming, I/O ports, and how the module is instantiated or used in other parts of the project.
5. Implement correct behavior based on design context. Do not guess randomly — reason it out.
6. Avoid vendor-specific features unless explicitly required.
7. Do not output comments, explanation, or anything outside the required code region unless explicitly asked.

Your output should contain **only the Verilog code** that fits inside the marked region.

/*
 * Copyright (c) 2024 Eric Fogleman
 * SPDX-License-Identifier: GPL-2.0-or-later
 *
 * Blocks for fully-segmented mismatch shaping encoder 
 * from SEGMENTED MISMATCH-SHAPING D/A CONVERSION, 
 * Fishov, Fogleman, Siragusa, Galton 
 * 2002 IEEE International Symposium on Circuits and Systems (ISCAS)
 */
/* verilator lint_off DECLFILENAME */

// retiming reg
module ef_smsdac_reg #(parameter BITS = 8) (
    input i_clk,
    input i_rst_b,
    input [BITS-1:0] i_d,
    output reg [BITS-1:0] o_q );

	always @( posedge i_clk, negedge i_rst_b ) 
		if ( i_rst_b == 1'b0 ) 
			o_q <= 0;
		else
			o_q <= i_d;

endmodule

// the segmenting switching block
module ef_smsdac_mse_seg_sb ( 
	input i_clk, 
	input i_rst_b, 
	input i_r, 
	input i_en, 
	input i_x, 
	input i_xc, 
	output wire [1:0] o_y, 
	output wire o_yc );

	wire odd = i_x ^ i_xc;	// input parity
	wire q;				// switching sequence

	// "carry" output (lsb weight); round up/down on odd input
	assign o_yc = odd ? q : i_x;

	// 3-level DAC outputs
	assign o_y[1] = odd & ~q;
	assign o_y[0] = ~odd | ~q;

	// switching sequence state machine
	ef_smsdac_mse_sb_sm sb_sm(
		.i_clk(i_clk),
		.i_rst_b(i_rst_b),
		.i_odd(odd),
		.i_r(i_r),
		.i_en(i_en),
		.o_q(q));
			 
endmodule

// the binary switching block
module ef_smsdac_mse_bin_sb ( 
	input i_clk,
	input i_rst_b,
	input i_r,
	input i_en,
	input i_x,
	input i_xc,
	output wire [1:0] o_y) ;

	wire q;				// switching sequence
	wire odd = i_x ^ i_xc;		// input parity

	// 3-level DAC output
	// if odd, split based on switching sequence
	assign o_y[1] = odd ? q : i_xc;
	assign o_y[0] = odd ? ~q : i_xc;

	// switching sequence state machine
	ef_smsdac_mse_sb_sm sb_sm(
		.i_clk(i_clk),
		.i_rst_b(i_rst_b),
		.i_odd(odd),
		.i_r(i_r),
		.i_en(i_en),
		.o_q(q));

endmodule

// the switching sequence state machine
// updates only when sb input is odd
// i_r is a random dither bit
// i_en == 1'b0 halts state machine and routes i_r to q
module ef_smsdac_mse_sb_sm( 
	input i_clk, 
	input i_rst_b, 
	input i_odd, 
	input i_r, 
	input i_en, 
	output wire o_q );

	// switching sequence; update only on odd inputs
	// en == 1'b0 stops mismatch shaping (static encoder)
	reg [1:0] q;
	wire [1:0] q_d;   // next state info
	assign q_d[1] = (i_en & i_odd) ? ~q[1]: q[1]; 
	assign q_d[0] = (i_en & i_odd) ? ( q[1] ? i_r : ~q[0] ) : q[0];

	// enabled:  highpass shaping
	// disabled:  random shaping (whitening)
	assign o_q = i_en ? q[0] : i_r;

	always @( posedge i_clk, negedge i_rst_b ) begin 
		if ( i_rst_b == 1'b0 ) 
			q <= 2'b00;
		else  
			q <= q_d;
	end

endmodule
/* verilator lint_on DECLFILENAME */


/*
 * Copyright (c) 2024 Eric Fogleman
 * SPDX-License-Identifier: Apache-2.0
  *
 * LFSR w/ multiple tapped outputs
 * updates K states at a time to avoid correlation in tapped bits
 * From: A 3.3-V SINGLE-POLY CMOS AUDIO ADC DELTA–SIGMA MODULATOR WITH 98-DB 
 * PEAK SINAD AND 105-DB PEAK SFDR, Fogleman et al
 */

/* verilator lint_off DECLFILENAME */
// n = 20, 1 + x^3 + x^n, K = 8
module ef_lfsr20_8 ( 
	input i_clk, 
	input i_rst_b, 
	input i_en, 
	output wire [7:0] o_r );
// >>> Module Implementation Begin

// <<< Module Implementation End
endmodule


/* verilator lint_on DECLFILENAME */


/*
 * Copyright (c) 2024 Eric Fogleman
 * SPDX-License-Identifier: GPL-2.0-or-later
 *
 * 8-b fully-segmented mismatch shaping encoder 
 * from SEGMENTED MISMATCH-SHAPING D/A CONVERSION, 
 * Fishov, Fogleman, Siragusa, Galton 
 * 2002 IEEE International Symposium on Circuits and Systems (ISCAS)
 * y0 drives 1x 3-level DAC
 * y1: 2x, y2: 4x, ..., y6: 64x
 * 128x DAC driven by x7, yc
 * i_en == 1'b0 stops mismatch shaping (static encoder)
 */
/* verilator lint_off DECLFILENAME */

// 8-b hierarchical encoder
// seg switching blocks to requantize the input
// binary switching blocks for each scaled 3-level DAC
module ef_smsdac8_mse ( 
	input i_clk,
	input i_rst_b,
	input i_en,
	input [7:0] i_x,
	input i_xc,
	input [7:0] i_r,
	output [13:0] o_y );	// y64[1:0], y32[1:0], ..., y1[1;0] 
 
	// 8 layers:  input 7, output 0
	// elements numbered top to bottom.  
	// signals and instances use layer, element notation:
	// u_s70 is 0th switching block in layer 7
	// y70 is output of layer 7, element 0

	// carry outs from segmenting switching blocks
	wire yc70, yc60, yc50, yc40, yc30, yc20, yc10;

	wire [1:0] y10, y20, y30, y40, y50, y60;  // to binary encoders
    // verilator lint_off UNUSEDSIGNAL
	wire [1:0] y70;	 // unused
    // verilator lint_on UNUSEDSIGNAL

	// layer 7
	ef_smsdac_mse_seg_sb u_s70( 
		.i_x(i_x[0]), 
		.i_xc(i_xc), 
		.i_r(i_r[7]), 
		.i_en(i_en),
		.i_clk(i_clk), 
		.i_rst_b(i_rst_b), 
		.o_yc(yc70), 
		.o_y(y70[1:0]) );  // unused

	// layer 6
	ef_smsdac_mse_seg_sb u_s60( 
		.i_x(i_x[1]), 
		.i_xc(yc70), 
		.i_r(i_r[6]), 
		.i_en(i_en),
		.i_clk(i_clk), 
		.i_rst_b(i_rst_b), 
		.o_yc(yc60), 
		.o_y(y60[1:0]) );

	ef_smsdac_mse_bin_sb u_s06(
		.i_x(y60[1]), 
		.i_xc(y60[0]),
		.i_r(i_r[6]), 
		.i_en(i_en),
		.i_clk(i_clk),
		.i_rst_b(i_rst_b),
		.o_y(o_y[1:0]) );	// 1*delta DAC

	// layer 5
	ef_smsdac_mse_seg_sb u_s50( 
		.i_x(i_x[2]), 
		.i_xc(yc60), 
		.i_r(i_r[5]),
		.i_en(i_en), 
		.i_clk(i_clk), 
		.i_rst_b(i_rst_b), 
		.o_yc(yc50), 
		.o_y(y50[1:0]) );

	ef_smsdac_mse_bin_sb u_s05(
		.i_x(y50[1]), 
		.i_xc(y50[0]),
		.i_r(i_r[5]), 
		.i_en(i_en),
		.i_clk(i_clk),
		.i_rst_b(i_rst_b),
		.o_y(o_y[3:2]) );	// 2*delta DAC

	// layer 4
	ef_smsdac_mse_seg_sb u_s40( 
		.i_x(i_x[3]), 
		.i_xc(yc50), 
		.i_r(i_r[4]),
		.i_en(i_en), 
		.i_clk(i_clk), 
		.i_rst_b(i_rst_b), 
		.o_yc(yc40), 
		.o_y(y40[1:0]) );

	ef_smsdac_mse_bin_sb u_s04(
		.i_x(y40[1]), 
		.i_xc(y40[0]),
		.i_r(i_r[4]), 
		.i_en(i_en),
		.i_clk(i_clk),
		.i_rst_b(i_rst_b),
		.o_y(o_y[5:4]) );	// 4*delta DAC

	// layer 3
	ef_smsdac_mse_seg_sb u_s30( 
		.i_x(i_x[4]), 
		.i_xc(yc40), 
		.i_r(i_r[3]),
		.i_en(i_en), 
		.i_clk(i_clk), 
		.i_rst_b(i_rst_b), 
		.o_yc(yc30), 
		.o_y(y30[1:0]) );
		
	ef_smsdac_mse_bin_sb u_s03(
		.i_x(y30[1]), 
		.i_xc(y30[0]),
		.i_r(i_r[3]), 
		.i_en(i_en),
		.i_clk(i_clk),
		.i_rst_b(i_rst_b),
		.o_y(o_y[7:6]) );	// 8*delta DAC
		
	// layer 2
	ef_smsdac_mse_seg_sb u_s20( 
		.i_x(i_x[5]), 
		.i_xc(yc30), 
		.i_r(i_r[2]),
		.i_en(i_en), 
		.i_clk(i_clk), 
		.i_rst_b(i_rst_b), 
		.o_yc(yc20), 
		.o_y(y20[1:0]) );
		
	ef_smsdac_mse_bin_sb u_s02(
		.i_x(y20[1]), 
		.i_xc(y20[0]),
		.i_r(i_r[2]), 
		.i_en(i_en),
		.i_clk(i_clk),
		.i_rst_b(i_rst_b),
		.o_y(o_y[9:8]) );	// 16*delta DAC

	// layer 1
	ef_smsdac_mse_seg_sb u_s10( 
		.i_x(i_x[6]), 
		.i_xc(yc20), 
		.i_r(i_r[1]),
		.i_en(i_en), 
		.i_clk(i_clk), 
		.i_rst_b(i_rst_b), 
		.o_yc(yc10), 
		.o_y(y10[1:0]) );
		
	ef_smsdac_mse_bin_sb u_s01(
		.i_x(y10[1]), 
		.i_xc(y10[0]),
		.i_r(i_r[1]), 
		.i_en(i_en),
		.i_clk(i_clk),
		.i_rst_b(i_rst_b),
		.o_y(o_y[11:10]) );	// 32*delta DAC
		
	// layer 0
	// {i_x[7], yc10};		
	// ms-segment from {input msb, final carry out}
	ef_smsdac_mse_bin_sb u_s00( 
		.i_x(i_x[7]), 
		.i_xc(yc10), 
		.i_r(i_r[0]),
		.i_en(i_en), 
		.i_clk(i_clk), 
		.i_rst_b(i_rst_b), 
		.o_y(o_y[13:12]) );	// 64*delta DAC

endmodule


// DAC top level 
//
module ef_smsdac8_top( 
    input i_clk, 
    input i_rst_b, 
    input i_en_enc, 
    input i_en_dith, 
    input [7:0] i_x, 
    output wire [13:0] o_y );	// y64[1:0], y32[1:0], ..., y1[1;0] 

    wire [7:0] x_sync1, x_sync2; // input data synchronizer
    wire [7:0] r;  // random dither bits
    wire [13:0] y;  // encoder outputs to final reclock

    // input data sync reg
    ef_smsdac_reg #(.BITS(8)) u_sync1(
        .i_clk(i_clk), 
        .i_rst_b(i_rst_b), 
        .i_d(i_x), 
        .o_q(x_sync1) );  

    // input data sync reg
    ef_smsdac_reg #(.BITS(8)) u_sync2(
        .i_clk(i_clk), 
        .i_rst_b(i_rst_b), 
        .i_d(x_sync1), 
        .o_q(x_sync2) );  

    // mismatch-shaping encoder
    ef_smsdac8_mse u_dac(
        .i_clk(i_clk), 
        .i_rst_b(i_rst_b), 
        .i_en(i_en_enc),
        .i_x(x_sync2), 
        .i_xc(1'b0),  // no carry in
        .i_r(r), 
        .o_y(y) );

    // LFSR
    ef_lfsr20_8 u_lfsr(
        .i_clk(i_clk),
        .i_rst_b(i_rst_b),
        .i_en(i_en_dith),
        .o_r(r) );

    // output retiming reg
    ef_smsdac_reg #(.BITS(14)) u_reg(
        .i_clk(i_clk), 
        .i_rst_b(i_rst_b), 
        .i_d(y), 
        .o_q(o_y) );

endmodule
/* verilator lint_on DECLFILENAME */


/*
 * Copyright (c) 2024 Eric Fogleman
 * SPDX-License-Identifier: Apache-2.0
 *
 * TT project wrapper for ef_smsdac
 * 1-10 MHz Segmented mismatch-shaping DAC 
 * Input clock at 1-50 MHz 
 * 8-b unsigned input data on ui_in[7:0]; sync'd to clk (oversampled)
 * uo_out[7:0] connect to {64x, 32x, 16x, 8x} weight 3-level DACs
 * uio_out[7:3] optionally connect to {4x, 2x, 1x} 3-level DACs 
 */

`default_nettype none

module tt_um_ejfogleman_smsdac8 (
    input  wire [7:0] ui_in,    // Dedicated inputs
    output wire [7:0] uo_out,   // Dedicated outputs
    // verilator lint_off UNUSEDSIGNAL
    input  wire [7:0] uio_in,   // IOs: Input path
    // verilator lint_off UNUSEDSIGNAL
    output wire [7:0] uio_out,  // IOs: Output path
    output wire [7:0] uio_oe,   // IOs: Enable path (active high: 0=input, 1=output)
    input  wire       ena,      // will go high when the design is enabled
    input  wire       clk,      // clock
    input  wire       rst_n     // reset_n - low to reset
    );
  
  // release reset when project enabled
  wire ena_and_rst_n = ena & rst_n;

  assign uio_oe[7:0] = 8'b11111100;    // uio[7:2] used to bring out extra DAC lsbs
  assign uio_out[1:0] = 2'h0;    // uio[1:0] used as control inputs

  ef_smsdac8_top u_top( 
    .i_clk(clk), 
    .i_rst_b(ena_and_rst_n), 
    .i_en_enc(uio_in[0]),
    .i_en_dith(uio_in[1]),
    .i_x(ui_in[7:0]), 
    .o_y({uo_out[7:0],uio_out[7:2]}) );

endmodule


Reference solution:
rtl/generated.v: 
	reg [19:0] q;
	assign o_r[7:0] = q[8:1];  

	always @( posedge i_clk, negedge i_rst_b ) begin
		if( ~i_rst_b ) begin
			q[19:1] <= 19'b0;
			q[0] <= 1'b1;
		end
		else begin
			if ( i_en ) begin
				// jumps 8 states per clock
				q[19:12] <= q[7:0] ^ q[10:3];
				q[11:0] <= q[19:8];
			end
		end
	end